Name the unknown
Write what the question asks for, including its unit. This prevents you from solving for a related but incorrect quantity.
215 formulas · 11 subjects · Always free
Math formula library with explanations and examples.
Search the core formulas used from middle school through high school, AP, and university mathematics. Every entry defines its symbols, states when it is valid, and shows a quick example.
Find the relationship—not just the equation.
A formula is a compact statement about how quantities are connected. Before substituting numbers, identify the unknown, match every symbol to a known value, and confirm that the formula’s conditions describe the problem you actually have.
Browse formulas by subject
Each subject has its own page, grouped by topic, and every formula has a page of its own with symbols, conditions, and a worked example.
How to use any formula without guessing
Match the symbols
List the given values beside the formula’s variables. Convert measurements so every unit belongs to one consistent system.
Check the conditions
Look for requirements such as a right angle, nonzero denominator, independent trials, convergence, or a particular angle mode.
Substitute and verify
Keep exact values during the work. At the end, check the sign, size, unit, domain, and original relationship.
215 formulas shown
Browse all 215 math formulas by subject
Add fractions
$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$
Rewrite two fractions over a common denominator, then add their numerators.
Symbols, conditions, and example ＋
What the symbols mean
$a,c$ are numerators and $b,d$ are nonzero denominators.
Use this formula when
$b\ne0$ and $d\ne0$; simplify the result when numerator and denominator share a factor.
Quick example
$\frac{2}{3}+\frac{1}{4}=\frac{8+3}{12}=\frac{11}{12}$.
Multiply fractions
$$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$$
Multiply numerators together and denominators together.
Symbols, conditions, and example ＋
What the symbols mean
$a,c$ are numerators and $b,d$ are denominators.
Use this formula when
$b\ne0$ and $d\ne0$; cancel common factors before or after multiplying.
Quick example
$\frac{3}{5}\cdot\frac{10}{9}=\frac{30}{45}=\frac{2}{3}$.
Divide fractions
$$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$$
Multiply by the reciprocal of the divisor.
Symbols, conditions, and example ＋
What the symbols mean
$c/d$ is the divisor; $d/c$ is its reciprocal.
Use this formula when
$b,c,d\ne0$ because neither a denominator nor the divisor may be zero.
Quick example
$\frac{4}{7}\div\frac{2}{3}=\frac{4}{7}\cdot\frac{3}{2}=\frac{6}{7}$.
Percent of a quantity
$$\text{part}=\frac{p}{100}\cdot\text{whole}$$
Find a stated percentage of a whole quantity.
Symbols, conditions, and example ＋
What the symbols mean
$p$ is the percent, part is the selected amount, and whole is the reference amount.
Use this formula when
The whole must represent the same unit and reference group as the part.
Quick example
$15\%$ of $80$ is $0.15(80)=12$.
Percent change
$$\%\text{ change}=\frac{\text{new}-\text{original}}{|\text{original}|}\cdot100\%$$
Compare a change with the magnitude of its starting value.
Symbols, conditions, and example ＋
What the symbols mean
new and original are measured in the same units.
Use this formula when
$\text{original}\ne0$; a positive result is an increase and a negative result is a decrease.
Quick example
From $50$ to $62$: $\frac{62-50}{50}(100\%)=24\%$ increase.
Distance, rate, and time
$$d=rt$$
Connect distance traveled, constant rate, and elapsed time.
Symbols, conditions, and example ＋
What the symbols mean
$d$ is distance, $r$ is rate, and $t$ is time.
Use this formula when
Use compatible units and a constant average rate over the interval.
Quick example
At $60\text{ km/h}$ for $2.5\text{ h}$, $d=60(2.5)=150\text{ km}$.
Simple interest
$$I=Prt,\qquad A=P(1+rt)$$
Calculate interest only on the original principal.
Symbols, conditions, and example ＋
What the symbols mean
$P$ is principal, $r$ is the decimal annual rate, $t$ is years, $I$ is interest, and $A$ is total.
Use this formula when
The rate and time unit must agree; this model does not compound interest.
Quick example
$P=1000$, $r=0.05$, $t=3$ gives $I=150$ and $A=1150$.
Compound interest
$$A=P\left(1+\frac{r}{n}\right)^{nt}$$
Grow an initial amount when interest is added repeatedly.
Symbols, conditions, and example ＋
What the symbols mean
$P$ is principal, $r$ is annual decimal rate, $n$ is compounds per year, and $t$ is years.
Use this formula when
$n>0$; the annual rate and time unit must agree.
Quick example
$1000$ at $6\%$ monthly for $2$ years gives $1000(1+0.06/12)^{24}\approx1127.16$.
Slope between two points
$$m=\frac{y_2-y_1}{x_2-x_1}$$
Measure vertical change per unit of horizontal change.
Symbols, conditions, and example ＋
What the symbols mean
$(x_1,y_1)$ and $(x_2,y_2)$ are points; $m$ is slope.
Use this formula when
$x_2\ne x_1$; a vertical line has undefined slope.
Quick example
Through $(1,2)$ and $(5,10)$, $m=(10-2)/(5-1)=2$.
Slope-intercept form
$$y=mx+b$$
Write a nonvertical line using its slope and vertical intercept.
Symbols, conditions, and example ＋
What the symbols mean
$m$ is slope and $b$ is the $y$-intercept.
Use this formula when
This form does not directly represent a vertical line $x=c$.
Quick example
Slope $3$ and intercept $-2$ give $y=3x-2$.
Point-slope form
$$y-y_1=m(x-x_1)$$
Write a line from one point and its slope.
Symbols, conditions, and example ＋
What the symbols mean
$m$ is slope and $(x_1,y_1)$ is a point on the line.
Use this formula when
Use for nonvertical lines; check by substituting the known point.
Quick example
Slope $-2$ through $(3,5)$ gives $y-5=-2(x-3)$.
Standard form of a line
$$Ax+By=C$$
Represent horizontal, vertical, and slanted lines in one form.
Symbols, conditions, and example ＋
What the symbols mean
$A,B,C$ are constants and $A,B$ are not both zero.
Use this formula when
For integer standard form, clear fractions and usually choose $A\ge0$.
Quick example
$y=2x+3$ becomes $2x-y=-3$.
Quadratic formula
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Find every real or complex root of a quadratic equation.
Symbols, conditions, and example ＋
What the symbols mean
$a,b,c$ are coefficients of $ax^2+bx+c=0$.
Use this formula when
$a\ne0$; keep the entire numerator over $2a$.
Quick example
For $2x^2+3x+6=0$, $x=\frac{-3\pm i\sqrt{39}}{4}$.
Discriminant
$$\Delta=b^2-4ac$$
Predict the number and type of roots before solving a quadratic.
Symbols, conditions, and example ＋
What the symbols mean
$a,b,c$ come from $ax^2+bx+c=0$.
Use this formula when
$\Delta>0$ gives two real roots, $\Delta=0$ one repeated root, and $\Delta<0$ two complex roots.
Quick example
For $x^2-6x+9$, $\Delta=36-36=0$, so $x=3$ is repeated.
Quadratic vertex
$$x_v=-\frac{b}{2a},\qquad y_v=f(x_v)$$
Locate the turning point of a parabola in standard form.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ are coefficients of $f(x)=ax^2+bx+c$.
Use this formula when
$a\ne0$; the vertex is a minimum if $a>0$ and a maximum if $a<0$.
Quick example
For $f(x)=2x^2-8x+1$, $x_v=2$ and $y_v=-7$.
Difference of squares
$$a^2-b^2=(a-b)(a+b)$$
Factor a subtraction of two perfect squares.
Symbols, conditions, and example ＋
What the symbols mean
$a$ and $b$ are the square roots of the two terms.
Use this formula when
The middle operation must be subtraction; $a^2+b^2$ does not factor this way over the reals.
Quick example
$9x^2-25=(3x-5)(3x+5)$.
Sum and difference of cubes
$$a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$$
Factor a sum or difference of perfect cubes.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ are the cube roots of the original terms.
Use this formula when
The sign in the quadratic factor is opposite the first sign; its final term is positive.
Quick example
$x^3-8=(x-2)(x^2+2x+4)$.
Product and quotient powers
$$a^m a^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n}$$
Combine powers that have the same base.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is the shared base and $m,n$ are exponents.
Use this formula when
For the quotient, $a\ne0$; do not combine exponents when bases differ.
Quick example
$x^5/x^2=x^{5-2}=x^3$ for $x\ne0$.
Power of a power
$$(a^m)^n=a^{mn}$$
Raise an existing power by multiplying exponents.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is the base and $m,n$ are exponents.
Use this formula when
A power of a product distributes, but a power of a sum generally does not.
Quick example
$(x^3)^4=x^{12}$.
Rational exponent
$$a^{m/n}=\sqrt[n]{a^m}$$
Translate between radical notation and fractional exponents.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is the root index and $m$ is the power.
Use this formula when
Over the reals, if $n$ is even then $a\ge0$; $n\ne0$.
Quick example
$27^{2/3}=(\sqrt[3]{27})^2=9$.
Arithmetic sequence term
$$a_n=a_1+(n-1)d$$
Find a term in a sequence with constant difference.
Symbols, conditions, and example ＋
What the symbols mean
$a_1$ is the first term, $d$ the common difference, and $n$ the term number.
Use this formula when
$n$ is a positive integer and the difference must be constant.
Quick example
If $a_1=4$ and $d=3$, then $a_{10}=4+9(3)=31$.
Arithmetic series sum
$$S_n=\frac{n}{2}(a_1+a_n)=\frac{n}{2}\left[2a_1+(n-1)d\right]$$
Add the first $n$ terms of an arithmetic sequence.
Symbols, conditions, and example ＋
What the symbols mean
$S_n$ is the sum and $a_1,a_n$ are the first and last terms.
Use this formula when
$n$ is a positive integer and terms must have constant difference.
Quick example
$2+5+8+11+14=\frac{5}{2}(2+14)=40$.
Geometric sequence term
$$a_n=a_1r^{n-1}$$
Find a term in a sequence with constant ratio.
Symbols, conditions, and example ＋
What the symbols mean
$a_1$ is the first term, $r$ the common ratio, and $n$ the term number.
Use this formula when
$n$ is a positive integer; when dividing to find $r$, the earlier term must be nonzero.
Quick example
If $a_1=3$ and $r=2$, then $a_6=3(2^5)=96$.
Finite geometric series
$$S_n=a_1\frac{1-r^n}{1-r}$$
Add the first $n$ terms of a geometric sequence.
Symbols, conditions, and example ＋
What the symbols mean
$a_1$ is the first term, $r$ the common ratio, and $n$ the number of terms.
Use this formula when
$r\ne1$; if $r=1$, then $S_n=na_1$.
Quick example
$1+2+4+8=1(1-2^4)/(1-2)=15$.
Logarithm definition
$$\log_b x=y\iff b^y=x$$
Translate between logarithmic and exponential statements.
Symbols, conditions, and example ＋
What the symbols mean
$b$ is the base, $x$ the positive argument, and $y$ the exponent.
Use this formula when
$b>0$, $b\ne1$, and $x>0$.
Quick example
$\log_2 32=5$ because $2^5=32$.
Log product and quotient rules
$$\log_b(MN)=\log_bM+\log_bN,\quad\log_b\frac{M}{N}=\log_bM-\log_bN$$
Expand or combine logarithms of products and quotients.
Symbols, conditions, and example ＋
What the symbols mean
$M,N$ are logarithm arguments and $b$ is the base.
Use this formula when
$M,N>0$, $b>0$, and $b\ne1$.
Quick example
$\log_3(9x)=2+\log_3x$ when $x>0$.
Change of base
$$\log_b x=\frac{\ln x}{\ln b}=\frac{\log_a x}{\log_a b}$$
Evaluate a logarithm using a different available base.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is any valid new base and $b$ is the original base.
Use this formula when
$x>0$; $a,b>0$ and neither base equals $1$.
Quick example
$\log_2 10=\ln(10)/\ln(2)\approx3.3219$.
Exponential growth and decay
$$A=A_0e^{kt}$$
Model continuous proportional growth or decay.
Symbols, conditions, and example ＋
What the symbols mean
$A_0$ is the initial amount, $k$ the continuous rate, and $t$ time.
Use this formula when
Use consistent time units; $k>0$ models growth and $k<0$ decay.
Quick example
With $A_0=200$ and $k=0.04$, $A(5)=200e^{0.2}\approx244.28$.
Pythagorean theorem
$$a^2+b^2=c^2$$
Relate the legs and hypotenuse of a right triangle.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ are perpendicular legs and $c$ is the hypotenuse.
Use this formula when
The triangle must be right and $c$ must be opposite the right angle.
Quick example
Legs $6$ and $8$ give $c=\sqrt{36+64}=10$.
Distance in the plane
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Find straight-line distance between two coordinate points.
Symbols, conditions, and example ＋
What the symbols mean
$(x_1,y_1)$ and $(x_2,y_2)$ are endpoints.
Use this formula when
Both points must use the same coordinate scale and units.
Quick example
From $(1,2)$ to $(4,6)$, $d=\sqrt{3^2+4^2}=5$.
Midpoint
$$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$
Find the point halfway between two coordinate points.
Symbols, conditions, and example ＋
What the symbols mean
The endpoint coordinates are averaged component by component.
Use this formula when
Both points must be expressed in the same coordinate system.
Quick example
The midpoint of $(-2,3)$ and $(6,7)$ is $(2,5)$.
Triangle area
$$A=\frac12 bh$$
Find triangle area from a base and perpendicular height.
Symbols, conditions, and example ＋
What the symbols mean
$b$ is a chosen base and $h$ is its perpendicular height.
Use this formula when
$b,h\ge0$ and height must meet the base line at $90^\circ$.
Quick example
With $b=12$ and $h=7$, $A=42$ square units.
Heron’s formula
$$s=\frac{a+b+c}{2},\qquad A=\sqrt{s(s-a)(s-b)(s-c)}$$
Find triangle area from three side lengths.
Symbols, conditions, and example ＋
What the symbols mean
$a,b,c$ are side lengths and $s$ is semiperimeter.
Use this formula when
Positive side lengths must satisfy the triangle inequalities.
Quick example
Sides $3,4,5$ give $s=6$ and $A=\sqrt{6\cdot3\cdot2\cdot1}=6$.
Rectangle area and perimeter
$$A=lw,\qquad P=2l+2w$$
Measure the interior and boundary of a rectangle.
Symbols, conditions, and example ＋
What the symbols mean
$l$ is length and $w$ is width.
Use this formula when
$l,w\ge0$ and adjacent sides are perpendicular.
Quick example
For $l=8,w=3$, $A=24$ and $P=22$.
Parallelogram area
$$A=bh$$
Find area using one side as a base and its perpendicular height.
Symbols, conditions, and example ＋
What the symbols mean
$b$ is base length and $h$ is perpendicular height.
Use this formula when
Do not use a slanted side as height unless it is perpendicular to the base.
Quick example
Base $9$ and height $4$ give $A=36$.
Trapezoid area
$$A=\frac12(b_1+b_2)h$$
Multiply height by the average of the parallel bases.
Symbols, conditions, and example ＋
What the symbols mean
$b_1,b_2$ are parallel side lengths and $h$ is perpendicular distance between them.
Use this formula when
The identified bases must be parallel.
Quick example
Bases $5,11$ and height $3$ give $A=24$.
Circle circumference
$$C=2\pi r=\pi d$$
Measure the distance around a circle.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radius and $d=2r$ is diameter.
Use this formula when
$r\ge0$; use one consistent length unit.
Quick example
For $r=4$, $C=8\pi\approx25.13$.
Circle area
$$A=\pi r^2$$
Measure the region enclosed by a circle.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is the radius.
Use this formula when
$r\ge0$; the answer uses square units.
Quick example
For $r=4$, $A=16\pi\approx50.27$.
Circle equation
$$(x-h)^2+(y-k)^2=r^2$$
Describe all points a fixed distance from a center.
Symbols, conditions, and example ＋
What the symbols mean
$(h,k)$ is the center and $r$ is radius.
Use this formula when
$r\ge0$; both coordinate axes must use the same scale.
Quick example
Center $(2,-1)$ and radius $3$ give $(x-2)^2+(y+1)^2=9$.
Arc length
$$s=r\theta$$
Find the length cut from a circle by a central angle.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radius and $\theta$ is the central angle in radians.
Use this formula when
$\theta$ must be in radians; for degrees use $s=(\theta/360^\circ)2\pi r$.
Quick example
For $r=6$ and $\theta=\pi/3$, $s=2\pi$.
Sector area
$$A=\frac12r^2\theta$$
Find the area of a circular sector.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radius and $\theta$ is the central angle in radians.
Use this formula when
$\theta$ must be in radians; for degrees use $A=(\theta/360^\circ)\pi r^2$.
Quick example
For $r=3$ and $\theta=\pi/2$, $A=9\pi/4$.
Interior-angle sum
$$S=(n-2)180^\circ$$
Add all interior angles of an $n$-sided polygon.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is the number of sides.
Use this formula when
$n$ is an integer with $n\ge3$; the polygon is simple.
Quick example
A hexagon has sum $(6-2)180^\circ=720^\circ$.
Regular polygon area
$$A=\frac12aP=\frac{ns^2}{4\tan(\pi/n)}$$
Find area from apothem and perimeter or side count and side length.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is apothem, $P$ perimeter, $n$ side count, and $s$ side length.
Use this formula when
The polygon must be regular and $n\ge3$.
Quick example
A regular hexagon with side $2$ has $A=6(4)/(4\tan(\pi/6))=6\sqrt3$.
Prism volume
$$V=Bh$$
Find volume from a constant base cross-section.
Symbols, conditions, and example ＋
What the symbols mean
$B$ is base area and $h$ is perpendicular prism height.
Use this formula when
The solid must be a prism and units must agree.
Quick example
Base area $12$ and height $5$ give $V=60$ cubic units.
Cylinder volume and surface area
$$V=\pi r^2h,\qquad S=2\pi r^2+2\pi rh$$
Measure a right circular cylinder’s volume and total exterior area.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radius and $h$ is perpendicular height.
Use this formula when
$r,h\ge0$; total surface area includes both circular bases.
Quick example
For $r=2,h=5$, $V=20\pi$ and $S=28\pi$.
Pyramid volume
$$V=\frac13Bh$$
Find a pyramid’s volume from base area and perpendicular height.
Symbols, conditions, and example ＋
What the symbols mean
$B$ is base area and $h$ is perpendicular height.
Use this formula when
Use perpendicular height, not slant height.
Quick example
Base area $36$ and height $10$ give $V=120$.
Cone volume and surface area
$$V=\frac13\pi r^2h,\qquad S=\pi r^2+\pi r\ell$$
Measure a right circular cone’s volume and total surface area.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radius, $h$ vertical height, and $\ell$ slant height.
Use this formula when
$\ell=\sqrt{r^2+h^2}$ for a right cone; include the base for total area.
Quick example
For $r=3,h=4,\ell=5$, $V=12\pi$ and $S=24\pi$.
Sphere volume and surface area
$$V=\frac43\pi r^3,\qquad S=4\pi r^2$$
Measure a sphere’s enclosed volume and outer area.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radius.
Use this formula when
$r\ge0$; volume uses cubic units and surface area square units.
Quick example
For $r=3$, $V=36\pi$ and $S=36\pi$.
Right-triangle ratios
$$\sin\theta=\frac{\text{opp}}{\text{hyp}},\quad\cos\theta=\frac{\text{adj}}{\text{hyp}},\quad\tan\theta=\frac{\text{opp}}{\text{adj}}$$
Connect an acute angle to ratios of right-triangle sides.
Symbols, conditions, and example ＋
What the symbols mean
opp and adj are relative to $\theta$; hyp is opposite the right angle.
Use this formula when
Use a right triangle and a consistent angle mode.
Quick example
With opposite $3$ and hypotenuse $5$, $\sin\theta=3/5$.
Reciprocal identities
$$\csc\theta=\frac1{\sin\theta},\quad\sec\theta=\frac1{\cos\theta},\quad\cot\theta=\frac1{\tan\theta}$$
Define cosecant, secant, and cotangent as reciprocals.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is an angle.
Use this formula when
The denominator function must be nonzero.
Quick example
If $\cos\theta=2/3$, then $\sec\theta=3/2$.
Pythagorean identities
$$\sin^2\theta+\cos^2\theta=1,\quad1+\tan^2\theta=\sec^2\theta,\quad1+\cot^2\theta=\csc^2\theta$$
Relate squared trigonometric functions through the unit circle.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is an angle where the displayed functions exist.
Use this formula when
When taking square roots, choose sign from the angle’s quadrant.
Quick example
If $\sin\theta=3/5$ in quadrant I, then $\cos\theta=4/5$.
Sine sum and difference
$$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$$
Find sine of a combined angle.
Symbols, conditions, and example ＋
What the symbols mean
$A,B$ are angles in the same unit.
Use this formula when
Keep the same sign as the angle operation.
Quick example
$\sin75^\circ=\sin(45^\circ+30^\circ)=\frac{\sqrt6+\sqrt2}{4}$.
Cosine sum and difference
$$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$
Find cosine of a combined angle.
Symbols, conditions, and example ＋
What the symbols mean
$A,B$ are angles in the same unit.
Use this formula when
Cosine uses the opposite sign in the middle.
Quick example
$\cos75^\circ=\frac{\sqrt6-\sqrt2}{4}$.
Tangent sum and difference
$$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$
Find tangent of a combined angle.
Symbols, conditions, and example ＋
What the symbols mean
$A,B$ are angles.
Use this formula when
The component tangents and final denominator must be defined and nonzero.
Quick example
$\tan75^\circ=(1+1/\sqrt3)/(1-1/\sqrt3)=2+\sqrt3$.
Double-angle identities
$$\sin2\theta=2\sin\theta\cos\theta,\quad\cos2\theta=\cos^2\theta-\sin^2\theta$$
Rewrite trigonometric functions of twice an angle.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is an angle.
Use this formula when
Equivalent cosine forms include $1-2\sin^2\theta$ and $2\cos^2\theta-1$.
Quick example
If $\sin\theta=3/5$ and $\cos\theta=4/5$, then $\sin2\theta=24/25$.
Half-angle identities
$$\sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2},\quad\cos^2\frac{\theta}{2}=\frac{1+\cos\theta}{2}$$
Connect half angles to cosine of the original angle.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is an angle.
Use this formula when
Choose the sign of sine or cosine from the quadrant of $\theta/2$.
Quick example
$\sin^2(30^\circ)=(1-\cos60^\circ)/2=1/4$.
Law of sines
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
Solve a non-right triangle using opposite side-angle pairs.
Symbols, conditions, and example ＋
What the symbols mean
Sides $a,b,c$ lie opposite angles $A,B,C$.
Use this formula when
An SSA setup may have zero, one, or two valid triangles; angles sum to $180^\circ$.
Quick example
If $A=30^\circ$, $a=5$, $B=45^\circ$, then $b=5\sin45^\circ/\sin30^\circ=5\sqrt2$.
Law of cosines
$$c^2=a^2+b^2-2ab\cos C$$
Solve a triangle from SSS or SAS information.
Symbols, conditions, and example ＋
What the symbols mean
$C$ is included between sides $a,b$ and opposite side $c$.
Use this formula when
Match each angle with its opposite side; the Pythagorean theorem is the $C=90^\circ$ case.
Quick example
For $a=5,b=7,C=60^\circ$, $c=\sqrt{25+49-35}=\sqrt{39}$.
Triangle area with sine
$$A=\frac12ab\sin C$$
Find area from two sides and their included angle.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ enclose angle $C$.
Use this formula when
The angle must be the included angle between the chosen sides.
Quick example
Sides $8,10$ with included angle $30^\circ$ give $A=20$.
Degrees and radians
$$\theta_{\rm rad}=\theta_{\rm deg}\frac{\pi}{180},\qquad\theta_{\rm deg}=\theta_{\rm rad}\frac{180}{\pi}$$
Convert the two common angle units.
Symbols, conditions, and example ＋
What the symbols mean
The subscripts identify the angle unit.
Use this formula when
Keep $\pi$ for an exact radian answer.
Quick example
$150^\circ=150(\pi/180)=5\pi/6$.
Sinusoidal model
$$y=A\sin(B(x-C))+D$$
Model periodic behavior with amplitude, period, shift, and midline.
Symbols, conditions, and example ＋
What the symbols mean
$|A|$ is amplitude, $2\pi/|B|$ is period, $C$ is phase shift, and $D$ is midline.
Use this formula when
$B\ne0$; use radians unless the variable is explicitly measured in degrees.
Quick example
$y=3\sin(2x)+1$ has amplitude $3$, period $\pi$, and midline $y=1$.
Parabola standard forms
$$(x-h)^2=4p(y-k),\qquad(y-k)^2=4p(x-h)$$
Describe a parabola by vertex and focal distance.
Symbols, conditions, and example ＋
What the symbols mean
$(h,k)$ is vertex and $p$ is directed vertex-to-focus distance.
Use this formula when
$p\ne0$; the focus lies $p$ units along the axis and directrix lies $p$ units opposite.
Quick example
$(x-1)^2=8(y+2)$ has vertex $(1,-2)$ and focus $(1,0)$.
Ellipse standard form
$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$
Describe an axis-aligned ellipse centered at $(h,k)$.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ are positive semiaxis lengths; the larger denominator gives the major axis.
Use this formula when
$a,b>0$; if $a>b$, focal distance satisfies $c^2=a^2-b^2$.
Quick example
$x^2/25+y^2/9=1$ has vertices $(\pm5,0)$ and foci $(\pm4,0)$.
Hyperbola standard form
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$
Describe a horizontal axis-aligned hyperbola.
Symbols, conditions, and example ＋
What the symbols mean
$(h,k)$ is center; $a,b$ set vertex and asymptote scales.
Use this formula when
$a,b>0$; $c^2=a^2+b^2$ and asymptotes are $y-k=\pm(b/a)(x-h)$.
Quick example
$x^2/9-y^2/16=1$ has vertices $(\pm3,0)$ and asymptotes $y=\pm4x/3$.
Infinite geometric series
$$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}$$
Add infinitely many terms of a convergent geometric sequence.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is the first term and $r$ is the common ratio.
Use this formula when
Converges only when $|r|<1$.
Quick example
$1+\frac12+\frac14+\cdots=1/(1-1/2)=2$.
Complex modulus
$$|a+bi|=\sqrt{a^2+b^2}$$
Measure a complex number’s distance from the origin.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is real part and $b$ is imaginary coefficient.
Use this formula when
$a,b$ are real; modulus is always nonnegative.
Quick example
$|3-4i|=\sqrt{9+16}=5$.
Derivative definition
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$
Define instantaneous rate of change as a limit of secant slopes.
Symbols, conditions, and example ＋
What the symbols mean
$h$ is an input change and $f'(x)$ is tangent slope.
Use this formula when
The limit must exist and be finite for differentiability at $x$.
Quick example
For $f(x)=x^2$, the quotient becomes $2x+h$, so $f'(x)=2x$.
Power rule
$$\frac{d}{dx}x^n=nx^{n-1}$$
Differentiate a power of the variable.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is a constant exponent.
Use this formula when
Valid where $x^n$ is differentiable; domain restrictions still apply for fractional or negative powers.
Quick example
$\frac{d}{dx}x^5=5x^4$.
Product rule
$$(fg)'=f'g+fg'$$
Differentiate a product without treating it as two separate derivatives.
Symbols, conditions, and example ＋
What the symbols mean
$f,g$ are differentiable functions of the same variable.
Use this formula when
Both functions must be differentiable at the point.
Quick example
$\frac{d}{dx}(x^2e^x)=2xe^x+x^2e^x$.
Quotient rule
$$\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}$$
Differentiate a quotient of two functions.
Symbols, conditions, and example ＋
What the symbols mean
$f$ is numerator and $g$ denominator.
Use this formula when
$f,g$ must be differentiable and $g(x)\ne0$.
Quick example
$\frac{d}{dx}(x/e^x)=(e^x-xe^x)/e^{2x}=(1-x)e^{-x}$.
Chain rule
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$
Differentiate a composition from the outside inward.
Symbols, conditions, and example ＋
What the symbols mean
$g$ is the inner function and $f$ the outer function.
Use this formula when
Both required derivatives must exist.
Quick example
$\frac{d}{dx}(3x+1)^4=4(3x+1)^3(3)=12(3x+1)^3$.
Exponential and logarithmic derivatives
$$\frac{d}{dx}e^x=e^x,\quad\frac{d}{dx}a^x=a^x\ln a,\quad\frac{d}{dx}\ln x=\frac1x$$
Differentiate the fundamental exponential and logarithm functions.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is a positive constant base.
Use this formula when
$a>0$, $a\ne1$; for real $\ln x$, $x>0$.
Quick example
$\frac{d}{dx}5^x=5^x\ln5$.
Trigonometric derivatives
$$(\sin x)'=\cos x,\quad(\cos x)'=-\sin x,\quad(\tan x)'=\sec^2x$$
Differentiate sine, cosine, and tangent.
Symbols, conditions, and example ＋
What the symbols mean
$x$ is the angle input.
Use this formula when
These derivative formulas assume $x$ is measured in radians; tangent must be defined.
Quick example
$\frac{d}{dx}\sin(4x)=4\cos(4x)$ by the chain rule.
Inverse trigonometric derivatives
$$(\arcsin x)'=\frac1{\sqrt{1-x^2}},\quad(\arctan x)'=\frac1{1+x^2}$$
Differentiate two common inverse trigonometric functions.
Symbols, conditions, and example ＋
What the symbols mean
$x$ is a real input.
Use this formula when
For the arcsine derivative, $|x|<1$; arctangent is differentiable for all real $x$.
Quick example
$\frac{d}{dx}\arctan(2x)=2/(1+4x^2)$.
Linearization
$$L(x)=f(a)+f'(a)(x-a)$$
Approximate a differentiable function near a known point.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is the center and $L$ is the tangent-line approximation.
Use this formula when
$f$ must be differentiable near $a$; accuracy usually decreases farther from $a$.
Quick example
$\sqrt{4.1}\approx2+\frac14(0.1)=2.025$.
Power antiderivative
$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$$
Reverse the power rule to find an antiderivative.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is constant and $C$ represents every constant antiderivative.
Use this formula when
$n\ne-1$; when $n=-1$, use $\int dx/x=\ln|x|+C$.
Quick example
$\int x^3dx=x^4/4+C$.
Substitution rule
$$\int f(g(x))g'(x)\,dx=\int f(u)\,du$$
Reverse the chain rule by replacing an inner expression.
Symbols, conditions, and example ＋
What the symbols mean
$u=g(x)$ and $du=g'(x)dx$.
Use this formula when
The differential factor must be present up to a constant; transform bounds in a definite integral.
Quick example
$\int2x\cos(x^2)dx=\sin(x^2)+C$ using $u=x^2$.
Integration by parts
$$\int u\,dv=uv-\int v\,du$$
Reverse the product rule to integrate a product.
Symbols, conditions, and example ＋
What the symbols mean
$u$ is differentiated and $dv$ is integrated to obtain $v$.
Use this formula when
Choose $u$ so the remaining integral becomes simpler.
Quick example
$\int xe^x dx=xe^x-\int e^x dx=e^x(x-1)+C$.
Fundamental theorem of calculus
$$\int_a^b f(x)\,dx=F(b)-F(a),\qquad F'=f$$
Evaluate signed accumulated change using any antiderivative.
Symbols, conditions, and example ＋
What the symbols mean
$F$ is an antiderivative and $a,b$ are bounds.
Use this formula when
$f$ must satisfy the theorem’s integrability conditions, commonly continuity on $[a,b]$.
Quick example
$\int_0^2 3x^2dx=[x^3]_0^2=8$.
Accumulation derivative
$$\frac{d}{dx}\int_a^x f(t)\,dt=f(x)$$
Differentiate an accumulation function with a variable upper bound.
Symbols, conditions, and example ＋
What the symbols mean
$t$ is a dummy integration variable and $x$ is the upper limit.
Use this formula when
$f$ is continuous near $x$; apply the chain rule when the bound is $g(x)$.
Quick example
$\frac{d}{dx}\int_1^{x^2}\cos t\,dt=2x\cos(x^2)$.
Average value of a function
$$f_{\rm avg}=\frac1{b-a}\int_a^b f(x)\,dx$$
Find the constant height with the same signed area on an interval.
Symbols, conditions, and example ＋
What the symbols mean
$[a,b]$ is the interval.
Use this formula when
$a\ne b$ and $f$ must be integrable on the interval.
Quick example
For $f(x)=x^2$ on $[0,3]$, $f_{avg}=\frac13[x^3/3]_0^3=3$.
Area between curves
$$A=\int_a^b|f(x)-g(x)|\,dx$$
Add vertical distances between two curves.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ bound the region and $f,g$ are the curves.
Use this formula when
Split the integral where the top curve changes; absolute value ensures geometric area is nonnegative.
Quick example
Between $y=x$ and $y=x^2$ on $[0,1]$, $A=\int_0^1(x-x^2)dx=1/6$.
Arc length of a graph
$$L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx$$
Add infinitesimal straight-line lengths along $y=f(x)$.
Symbols, conditions, and example ＋
What the symbols mean
$f'$ is slope and $[a,b]$ is the horizontal interval.
Use this formula when
$f'$ should be continuous and the integral must converge.
Quick example
For $f(x)=0$ on $[0,5]$, $L=\int_0^5 1dx=5$.
Disk and washer volumes
$$V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)dx$$
Build a solid of revolution from circular cross-sections.
Symbols, conditions, and example ＋
What the symbols mean
$R$ is outer radius and $r$ inner radius.
Use this formula when
$R\ge r\ge0$ and radii are perpendicular distances to the axis of rotation.
Quick example
Rotating $y=x$ on $[0,1]$ about the $x$-axis gives $V=\pi\int_0^1x^2dx=\pi/3$.
Cylindrical shell volume
$$V=2\pi\int_a^b(\text{radius})(\text{height})\,dx$$
Build a solid of revolution from thin cylindrical shells.
Symbols, conditions, and example ＋
What the symbols mean
Radius is distance to the axis; height is top minus bottom.
Use this formula when
Use slices parallel to the axis and nonnegative geometric radius and height.
Quick example
Rotating the region under $y=x$ on $[0,1]$ about the $y$-axis gives $V=2\pi\int_0^1x^2dx=2\pi/3$.
Taylor series
$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$$
Represent a sufficiently regular function by derivatives at a center.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is center and $f^{(n)}(a)$ the order-$n$ derivative there.
Use this formula when
The Taylor series must converge to $f$ at the chosen $x$; smoothness alone is not always enough.
Quick example
$e^x=1+x+x^2/2!+x^3/3!+\cdots$ around $a=0$.
Taylor remainder bound
$$|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}$$
Bound the error after truncating a Taylor polynomial.
Symbols, conditions, and example ＋
What the symbols mean
$M$ bounds $|f^{(n+1)}|$ between $a$ and $x$.
Use this formula when
A valid finite bound $M$ must hold throughout the interval between center and input.
Quick example
For $e^x$ near $0$, choose $M=e^{|x|}$ to bound the next-order error.
Newton’s method
$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$
Improve an estimate of a root by following the tangent line to the axis.
Symbols, conditions, and example ＋
What the symbols mean
$x_n$ is the current estimate.
Use this formula when
$f'(x_n)\ne0$; convergence depends on the starting value and local behavior.
Quick example
For $x^2-2=0$ and $x_0=1.5$, $x_1=1.5-0.25/3\approx1.4167$.
Parametric derivatives
$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\qquad\frac{d^2y}{dx^2}=\frac{d}{dt}(dy/dx)\Big/\frac{dx}{dt}$$
Differentiate a curve whose coordinates depend on a parameter.
Symbols, conditions, and example ＋
What the symbols mean
$x=x(t)$ and $y=y(t)$.
Use this formula when
$dx/dt\ne0$ at the point for these quotients.
Quick example
If $x=t^2,y=t^3$, then $dy/dx=3t/2$ for $t\ne0$.
Polar area
$$A=\frac12\int_{\alpha}^{\beta}r(\theta)^2\,d\theta$$
Find area swept by a polar curve.
Symbols, conditions, and example ＋
What the symbols mean
$r(\theta)$ is radial distance and $[\alpha,\beta]$ is the angle interval.
Use this formula when
Trace the intended region once and split intervals if curves intersect or repeat.
Quick example
For $r=2$ and $0\le\theta\le\pi/2$, $A=\frac12\int_0^{\pi/2}4d\theta=\pi$.
Gradient
$$\nabla f=\left\langle f_x,f_y,f_z\right\rangle$$
Collect partial derivatives into the direction of steepest increase.
Symbols, conditions, and example ＋
What the symbols mean
$f_x,f_y,f_z$ are first partial derivatives.
Use this formula when
The partial derivatives should exist; differentiability is needed for the full geometric interpretation.
Quick example
For $f=x^2+y^2$, $\nabla f=\langle2x,2y\rangle$.
Directional derivative
$$D_{\mathbf u}f=\nabla f\cdot\mathbf u$$
Measure change per unit distance in a chosen direction.
Symbols, conditions, and example ＋
What the symbols mean
$\mathbf u$ is a unit direction vector.
Use this formula when
$f$ must be differentiable and $\|\mathbf u\|=1$.
Quick example
If $\nabla f=(2,3)$ and $\mathbf u=(1,0)$, then $D_{\mathbf u}f=2$.
Double and triple integrals
$$\iint_R f\,dA,\qquad\iiint_E f\,dV$$
Accumulate a density or function across a region or solid.
Symbols, conditions, and example ＋
What the symbols mean
$R$ is a planar region, $E$ a spatial region, and $dA,dV$ are area and volume elements.
Use this formula when
Choose bounds and Jacobian appropriate to the coordinate system; the integrals must converge.
Quick example
$\iint_{[0,1]^2}(x+y)dA=1$.
Line integral of a vector field
$$\int_C\mathbf F\cdot d\mathbf r=\int_a^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)dt$$
Accumulate the tangential component of a vector field along a curve.
Symbols, conditions, and example ＋
What the symbols mean
$C$ is oriented by parametrization $\mathbf r(t)$.
Use this formula when
The field and path should be sufficiently smooth; reversing orientation changes the sign.
Quick example
For $\mathbf F=(1,0)$ along $\mathbf r(t)=(t,t)$, $0\le t\le1$, the integral is $1$.
Arithmetic mean
$$\bar{x}=\frac1n\sum_{i=1}^n x_i$$
Find the balance-point average of observed values.
Symbols, conditions, and example ＋
What the symbols mean
$x_i$ are observations and $n$ is their count.
Use this formula when
$n>0$; the mean is sensitive to extreme values.
Quick example
For $2,4,9$, $\bar x=(2+4+9)/3=5$.
Weighted mean
$$\bar{x}_w=\frac{\sum w_ix_i}{\sum w_i}$$
Average values that contribute unequal weights.
Symbols, conditions, and example ＋
What the symbols mean
$x_i$ are values and $w_i$ their weights.
Use this formula when
$\sum w_i\ne0$; use nonnegative weights for the usual averaging interpretation.
Quick example
Scores $80,90$ weighted $1,2$ give $(80+180)/3\approx86.67$.
Population variance and standard deviation
$$\sigma^2=\frac1N\sum(x_i-\mu)^2,\qquad\sigma=\sqrt{\sigma^2}$$
Measure average squared distance from a population mean and return to original units.
Symbols, conditions, and example ＋
What the symbols mean
$N$ is population size and $\mu$ population mean.
Use this formula when
Use this denominator when the data are the complete population of interest.
Quick example
For population $1,3$, $\mu=2$, $\sigma^2=1$, and $\sigma=1$.
Sample variance and standard deviation
$$s^2=\frac1{n-1}\sum(x_i-\bar{x})^2,\qquad s=\sqrt{s^2}$$
Estimate population spread from a sample.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is sample size and $\bar x$ sample mean.
Use this formula when
$n>1$; $n-1$ corrects degrees of freedom after estimating the mean.
Quick example
For sample $1,3$, $\bar x=2$, $s^2=2$, and $s=\sqrt2$.
z-score
$$z=\frac{x-\mu}{\sigma}$$
Express a value as standard deviations above or below a mean.
Symbols, conditions, and example ＋
What the symbols mean
$x$ is value, $\mu$ population mean, and $\sigma$ population standard deviation.
Use this formula when
$\sigma>0$; a negative score lies below the mean.
Quick example
If $x=85,\mu=70,\sigma=10$, then $z=1.5$.
Sample covariance
$$s_{xy}=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{n-1}$$
Measure how two quantitative variables vary together.
Symbols, conditions, and example ＋
What the symbols mean
Paired values are $(x_i,y_i)$.
Use this formula when
$n>1$; covariance depends on measurement units.
Quick example
Positive covariance means larger $x$ values tend to occur with larger $y$ values.
Pearson correlation
$$r=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}}$$
Measure strength and direction of a linear sample relationship.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is unitless and lies between $-1$ and $1$.
Use this formula when
Both variables must vary; correlation does not establish causation and describes linear association.
Quick example
$r$ near $1$ indicates a strong positive linear association.
Least-squares regression line
$$\hat y=b_0+b_1x,\quad b_1=r\frac{s_y}{s_x},\quad b_0=\bar y-b_1\bar x$$
Predict a response using the line minimizing squared residuals.
Symbols, conditions, and example ＋
What the symbols mean
$b_1$ is slope, $b_0$ intercept, and $\hat y$ predicted response.
Use this formula when
$s_x>0$; interpret within the data range unless extrapolation is justified.
Quick example
If $r=.8,s_y=10,s_x=4$, then $b_1=2$.
Standard error of a mean
$$SE_{\bar x}=\frac{\sigma}{\sqrt n}\quad\text{or}\quad\frac{s}{\sqrt n}$$
Describe typical sample-mean variation across repeated samples.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is sample size and $\sigma$ or $s$ is spread.
Use this formula when
Observations should be independent; using $s$ estimates the unknown population standard deviation.
Quick example
If $s=12,n=36$, then $SE=2$.
Standard error of a proportion
$$SE_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$$
Describe sampling variation of a sample proportion.
Symbols, conditions, and example ＋
What the symbols mean
$p$ is population success proportion and $n$ sample size.
Use this formula when
Independence and a suitable random-sampling model are required; use $\hat p$ when estimating for a confidence interval.
Quick example
For $p=.5,n=100$, $SE=.05$.
One-mean confidence interval
$$\bar x\pm t^*\frac{s}{\sqrt n}$$
Estimate a population mean with a margin of error.
Symbols, conditions, and example ＋
What the symbols mean
$t^*$ comes from confidence level and $n-1$ degrees of freedom.
Use this formula when
Use independent random observations and check the sampling distribution is suitable for the sample size and shape.
Quick example
If $\bar x=50,s=10,n=25,t^*=2.064$, the interval is $50\pm4.128$.
One-proportion confidence interval
$$\hat p\pm z^*\sqrt{\frac{\hat p(1-\hat p)}{n}}$$
Estimate a population proportion.
Symbols, conditions, and example ＋
What the symbols mean
$\hat p$ is sample proportion and $z^*$ sets confidence level.
Use this formula when
Use random independent data and enough expected successes and failures for the normal approximation.
Quick example
If $\hat p=.60,n=100,z^*=1.96$, margin is about $.096$.
One-sample t statistic
$$t=\frac{\bar x-\mu_0}{s/\sqrt n}$$
Compare a sample mean with a null-hypothesis mean.
Symbols, conditions, and example ＋
What the symbols mean
$\mu_0$ is null value and $n-1$ is the degrees of freedom.
Use this formula when
Use independent data and verify the t-procedure’s shape or sample-size conditions.
Quick example
$\bar x=52,\mu_0=50,s=8,n=16$ gives $t=1$.
Chi-square statistic
$$\chi^2=\sum\frac{(O-E)^2}{E}$$
Compare observed categorical counts with counts expected under a null model.
Symbols, conditions, and example ＋
What the symbols mean
$O$ and $E$ are observed and expected cell counts.
Use this formula when
Counts must be independent and expected counts sufficiently large for the chi-square approximation.
Quick example
A cell with $O=12,E=10$ contributes $(12-10)^2/10=0.4$.
Residual
$$e=y-\hat y$$
Measure vertical prediction error for one observation.
Symbols, conditions, and example ＋
What the symbols mean
$y$ is observed response and $\hat y$ predicted response.
Use this formula when
Positive residual means the observation lies above the fitted model.
Quick example
Observed $18$ and predicted $15$ give residual $3$.
Complement rule
$$P(A^c)=1-P(A)$$
Find the probability that an event does not occur.
Symbols, conditions, and example ＋
What the symbols mean
$A^c$ is the complement of event $A$.
Use this formula when
$0\le P(A)\le1$ and $A,A^c$ partition the sample space.
Quick example
If rain has probability $.3$, no rain has probability $.7$.
Addition rule
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
Find the probability that at least one of two events occurs.
Symbols, conditions, and example ＋
What the symbols mean
$A\cup B$ means A or B; $A\cap B$ means both.
Use this formula when
Subtract the overlap once because it was counted twice.
Quick example
If $.4,.5,$ and overlap $.2$, then union probability is $.7$.
Conditional probability
$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$
Restrict probability to outcomes where $B$ occurred.
Symbols, conditions, and example ＋
What the symbols mean
$A\cap B$ is the joint event.
Use this formula when
$P(B)>0$.
Quick example
If $P(A\cap B)=.15$ and $P(B)=.3$, then $P(A\mid B)=.5$.
Multiplication rule
$$P(A\cap B)=P(A\mid B)P(B)$$
Find a joint probability using a conditional probability.
Symbols, conditions, and example ＋
What the symbols mean
$A\cap B$ means both events occur.
Use this formula when
For independent events this reduces to $P(A)P(B)$.
Quick example
If $P(B)=.4$ and $P(A\mid B)=.25$, then $P(A\cap B)=.10$.
Bayes’ theorem
$$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$
Reverse a conditional probability after observing evidence.
Symbols, conditions, and example ＋
What the symbols mean
$P(A)$ is prior and $P(A\mid B)$ posterior probability.
Use this formula when
$P(B)>0$; compute $P(B)$ across all relevant cases when not given.
Quick example
If $P(A)=.2,P(B\mid A)=.8,P(B)=.4$, then $P(A\mid B)=.4$.
Permutations
$$P(n,r)=\frac{n!}{(n-r)!}$$
Count ordered selections of $r$ distinct objects from $n$.
Symbols, conditions, and example ＋
What the symbols mean
$n!$ is factorial.
Use this formula when
$0\le r\le n$ are integers and order matters.
Quick example
Assigning 3 distinct roles among 5 people gives $5!/(5-3)!=60$.
Combinations
$$\binom nr=\frac{n!}{r!(n-r)!}$$
Count unordered selections of $r$ objects from $n$.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is population size and $r$ selection size.
Use this formula when
$0\le r\le n$ are integers and order does not matter.
Quick example
Choosing 3 of 5 people gives $\binom53=10$.
Binomial probability
$$P(X=k)=\binom nkp^k(1-p)^{n-k}$$
Find exactly $k$ successes in $n$ independent Bernoulli trials.
Symbols, conditions, and example ＋
What the symbols mean
$p$ is constant success probability.
Use this formula when
Fixed $n$, independent trials, two outcomes, and constant $p$.
Quick example
For $n=4,p=.5$, $P(X=2)=\binom42(.5)^4=6/16$.
Binomial mean and variance
$$E[X]=np,\qquad\operatorname{Var}(X)=np(1-p)$$
Summarize the center and spread of a binomial count.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is trial count and $p$ success probability.
Use this formula when
The variable must satisfy the binomial conditions.
Quick example
For $n=100,p=.2$, mean is $20$ and variance $16$.
Geometric probability
$$P(X=k)=(1-p)^{k-1}p$$
Find the trial number of the first success.
Symbols, conditions, and example ＋
What the symbols mean
$p$ is constant success probability and $k=1,2,\ldots$.
Use this formula when
Trials are independent with two outcomes and constant $p$.
Quick example
With $p=.25$, first success on trial 3 has probability $(.75)^2(.25)$.
Poisson probability
$$P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$$
Model event counts occurring at a constant average rate.
Symbols, conditions, and example ＋
What the symbols mean
$\lambda>0$ is expected count in the interval.
Use this formula when
Events should occur independently with approximately constant rate.
Quick example
If $\lambda=2$, $P(X=0)=e^{-2}$.
Expected value and variance
$$E[X]=\sum_x xP(X=x),\qquad\operatorname{Var}(X)=E[X^2]-E[X]^2$$
Find the long-run center and squared spread of a discrete random variable.
Symbols, conditions, and example ＋
What the symbols mean
$x$ ranges over possible values.
Use this formula when
Probabilities must be nonnegative and sum to $1$; the required sums must converge.
Quick example
For a fair die, $E[X]=(1+2+3+4+5+6)/6=3.5$.
Vector magnitude
$$\|\mathbf v\|=\sqrt{v_1^2+\cdots+v_n^2}$$
Measure Euclidean vector length.
Symbols, conditions, and example ＋
What the symbols mean
$v_i$ are vector components.
Use this formula when
Components must use compatible coordinate scales.
Quick example
$\|(3,4)\|=5$.
Unit vector
$$\widehat{\mathbf v}=\frac{\mathbf v}{\|\mathbf v\|}$$
Keep a vector’s direction while changing its length to one.
Symbols, conditions, and example ＋
What the symbols mean
$\mathbf v$ is the original vector.
Use this formula when
$\mathbf v\ne\mathbf0$.
Quick example
For $\mathbf v=(3,4)$, $\widehat{\mathbf v}=(3/5,4/5)$.
Dot product and angle
$$\mathbf u\cdot\mathbf v=\sum u_iv_i=\|\mathbf u\|\|\mathbf v\|\cos\theta$$
Measure alignment and recover the angle between vectors.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is the angle between nonzero vectors.
Use this formula when
For the angle formula both vectors must be nonzero; dot product zero means orthogonal.
Quick example
$(1,2)\cdot(3,4)=11$.
Vector projection
$$\operatorname{proj}_{\mathbf u}\mathbf v=\frac{\mathbf v\cdot\mathbf u}{\mathbf u\cdot\mathbf u}\mathbf u$$
Find the component of $\mathbf v$ pointing along $\mathbf u$.
Symbols, conditions, and example ＋
What the symbols mean
$\mathbf u$ supplies the projection direction.
Use this formula when
$\mathbf u\ne\mathbf0$.
Quick example
Project $(2,3)$ onto $(1,0)$ to get $(2,0)$.
Cross product
$$\mathbf u\times\mathbf v=\langle u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1\rangle$$
Produce a vector perpendicular to two three-dimensional vectors.
Symbols, conditions, and example ＋
What the symbols mean
$\mathbf u,\mathbf v\in\mathbb R^3$.
Use this formula when
Order matters: $\mathbf v\times\mathbf u=-(\mathbf u\times\mathbf v)$.
Quick example
$(1,0,0)\times(0,1,0)=(0,0,1)$.
Matrix multiplication
$$(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}$$
Compose two linear transformations by row-column products.
Symbols, conditions, and example ＋
What the symbols mean
$A$ is $m\times n$ and $B$ is $n\times p$.
Use this formula when
The inner dimensions must match; generally $AB\ne BA$.
Quick example
For row $(1,2)$ and column $(3,4)^T$, the entry is $1(3)+2(4)=11$.
Two-by-two determinant
$$\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc$$
Measure signed area scaling and test invertibility.
Symbols, conditions, and example ＋
What the symbols mean
$a,b,c,d$ are matrix entries.
Use this formula when
The matrix is invertible exactly when $ad-bc\ne0$.
Quick example
$\det\begin{bmatrix}2&1\\3&4\end{bmatrix}=8-3=5$.
Two-by-two inverse
$$A^{-1}=\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$$
Undo an invertible two-by-two linear transformation.
Symbols, conditions, and example ＋
What the symbols mean
$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$.
Use this formula when
$ad-bc\ne0$.
Quick example
For $A=\begin{bmatrix}2&1\\1&1\end{bmatrix}$, $A^{-1}=\begin{bmatrix}1&-1\\-1&2\end{bmatrix}$.
Inverse solution of a system
$$A\mathbf x=\mathbf b\implies\mathbf x=A^{-1}\mathbf b$$
Solve a square linear system using an inverse.
Symbols, conditions, and example ＋
What the symbols mean
$A$ is coefficient matrix, $\mathbf x$ unknown vector, and $\mathbf b$ constants.
Use this formula when
$A$ must be square and invertible; elimination is usually more efficient computationally.
Quick example
Verify a solution by multiplying $A\mathbf x$ and checking that it equals $\mathbf b$.
Eigenvalue equation
$$A\mathbf v=\lambda\mathbf v$$
Identify directions a transformation only scales.
Symbols, conditions, and example ＋
What the symbols mean
$\lambda$ is an eigenvalue and $\mathbf v$ its eigenvector.
Use this formula when
$\mathbf v\ne\mathbf0$.
Quick example
For diagonal $A=\operatorname{diag}(2,3)$, $(1,0)$ has eigenvalue $2$.
Characteristic equation
$$\det(A-\lambda I)=0$$
Find eigenvalues as roots of a polynomial.
Symbols, conditions, and example ＋
What the symbols mean
$I$ is identity matrix and $\lambda$ is the unknown eigenvalue.
Use this formula when
$A$ must be square.
Quick example
For $A=\operatorname{diag}(2,3)$, $(2-\lambda)(3-\lambda)=0$.
Rank-nullity theorem
$$\operatorname{rank}(A)+\operatorname{nullity}(A)=n$$
Split the input dimension into visible output directions and null-space directions.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is number of columns of $A$.
Use this formula when
$A$ represents a linear map from an $n$-dimensional domain.
Quick example
A $3$-column matrix of rank $2$ has nullity $1$.
Normal equations
$$A^TA\widehat{\mathbf x}=A^T\mathbf b$$
Find a vector whose model prediction is closest in squared distance to data.
Symbols, conditions, and example ＋
What the symbols mean
$\widehat{\mathbf x}$ is the least-squares solution.
Use this formula when
A unique solution is guaranteed when columns of $A$ are linearly independent; QR is often numerically safer.
Quick example
The residual $\mathbf b-A\widehat{\mathbf x}$ is orthogonal to every column of $A$.
Gram–Schmidt step
$$\mathbf u_k=\mathbf v_k-\sum_{j<k}\operatorname{proj}_{\mathbf u_j}\mathbf v_k$$
Remove earlier vector directions to build an orthogonal basis.
Symbols, conditions, and example ＋
What the symbols mean
$\mathbf v_k$ are independent input vectors and $\mathbf u_j$ orthogonal outputs.
Use this formula when
The starting vectors must be linearly independent to avoid a zero denominator during normalization.
Quick example
From $(1,0)$ and $(1,1)$, the second orthogonal vector becomes $(0,1)$.
Separable equation
$$\frac{dy}{dx}=g(x)h(y)\implies\int\frac{dy}{h(y)}=\int g(x)dx$$
Separate variables so each side can be integrated.
Symbols, conditions, and example ＋
What the symbols mean
$g$ depends only on $x$ and $h$ only on $y$.
Use this formula when
Check solutions lost when dividing by $h(y)=0$ and apply the initial condition after integrating.
Quick example
$y'=xy$ gives $\ln|y|=x^2/2+C$, so $y=Ce^{x^2/2}$.
Linear integrating factor
$$y'+p(x)y=q(x),\quad\mu(x)=e^{\int p(x)dx}$$
Turn a first-order linear equation into one product derivative.
Symbols, conditions, and example ＋
What the symbols mean
$\mu$ is the integrating factor.
Use this formula when
Write the equation with coefficient $1$ on $y'$ before computing $\mu$.
Quick example
After multiplication, $(\mu y)'=\mu q$, then integrate and solve for $y$.
Logistic growth
$$\frac{dP}{dt}=rP\left(1-\frac PK\right),\qquad P(t)=\frac{K}{1+Ae^{-rt}}$$
Model growth that slows near a carrying capacity.
Symbols, conditions, and example ＋
What the symbols mean
$K$ is carrying capacity, $r$ intrinsic rate, and $A=(K-P_0)/P_0$.
Use this formula when
$K,r,P_0>0$ for the usual population model.
Quick example
At $P=K/2$, growth rate is $rK/4$, its maximum.
Constant-coefficient characteristic equation
$$ay''+by'+cy=0\implies ar^2+br+c=0$$
Turn a homogeneous linear differential equation into an algebraic root problem.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is the characteristic root.
Use this formula when
$a\ne0$; solution form depends on distinct, repeated, or complex roots.
Quick example
$y''-3y'+2y=0$ has roots $1,2$, so $y=C_1e^x+C_2e^{2x}$.
Euler’s method
$$y_{n+1}=y_n+h f(x_n,y_n),\qquad x_{n+1}=x_n+h$$
Step along a differential equation using the current tangent slope.
Symbols, conditions, and example ＋
What the symbols mean
$h$ is step size and $y'=f(x,y)$.
Use this formula when
Smaller $h$ usually improves accuracy but increases work; accumulated error must be monitored.
Quick example
For $y'=y,y(0)=1,h=.1$, $y_1=1+.1(1)=1.1$.
Derivative transform
$$\mathcal L\{f'(t)\}=sF(s)-f(0)$$
Convert a time derivative into algebra while preserving an initial value.
Symbols, conditions, and example ＋
What the symbols mean
$F(s)=\mathcal L\{f(t)\}$.
Use this formula when
$f$ must satisfy conditions ensuring its Laplace transform exists.
Quick example
This turns many initial-value differential equations into equations for $F(s)$.
De Morgan’s laws
$$\neg(P\land Q)\equiv\neg P\lor\neg Q,\qquad\neg(P\lor Q)\equiv\neg P\land\neg Q$$
Move a negation through AND or OR while switching the connective.
Symbols, conditions, and example ＋
What the symbols mean
$P,Q$ are propositions.
Use this formula when
Apply the negation to every component, including quantified statements.
Quick example
“Not both A and B” means “not A or not B.”
Two-set inclusion–exclusion
$$|A\cup B|=|A|+|B|-|A\cap B|$$
Count a union without double-counting the overlap.
Symbols, conditions, and example ＋
What the symbols mean
$|A|$ means the number of elements in set $A$.
Use this formula when
Sets must be finite for this cardinality form.
Quick example
If $|A|=20,|B|=15,|A\cap B|=5$, then $|A\cup B|=30$.
Sum of first integers
$$\sum_{k=1}^{n}k=\frac{n(n+1)}2$$
Add consecutive positive integers efficiently.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is a nonnegative integer.
Use this formula when
For an empty sum at $n=0$, the result is $0$.
Quick example
$1+2+\cdots+100=100(101)/2=5050$.
Sum of squares
$$\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}6$$
Add squares of the first $n$ positive integers.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is a nonnegative integer.
Use this formula when
Use integer $n\ge0$.
Quick example
$1^2+2^2+3^2=3(4)(7)/6=14$.
Handshaking lemma
$$\sum_{v\in V}\deg(v)=2|E|$$
Relate all vertex degrees to the number of edges in an undirected graph.
Symbols, conditions, and example ＋
What the symbols mean
$V$ is vertex set and $E$ edge set.
Use this formula when
Each undirected edge contributes two degree incidences; loops require the usual degree-two convention.
Quick example
A graph with $7$ edges has total degree $14$.
Euler formula for planar graphs
$$|V|-|E|+|F|=2$$
Relate vertices, edges, and faces in a connected planar embedding.
Symbols, conditions, and example ＋
What the symbols mean
$F$ includes the unbounded outer face.
Use this formula when
The graph must be connected and planar; for $c$ components use $|V|-|E|+|F|=1+c$.
Quick example
A triangle has $3-3+2=2$.
Euclidean algorithm
$$\gcd(a,b)=\gcd(b,a\bmod b)$$
Reduce a greatest-common-divisor problem to smaller remainders.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ are integers, not both zero.
Use this formula when
Repeat until the remainder is zero; the last nonzero remainder is the positive gcd.
Quick example
$\gcd(48,18)=\gcd(18,12)=\gcd(12,6)=6$.
Modular congruence
$$a\equiv b\pmod n\iff n\mid(a-b)$$
State that two integers have the same remainder modulo $n$.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is the modulus.
Use this formula when
$n$ is a positive integer in the standard convention.
Quick example
$17\equiv5\pmod{12}$ because $12\mid(17-5)$.
Equivalent proportions
$$\frac{a}{b}=\frac{c}{d}\iff ad=bc$$
Test or solve an equality between two ratios by cross multiplication.
Symbols, conditions, and example ＋
What the symbols mean
$a,b,c,d$ are quantities arranged in matching order.
Use this formula when
$b\ne0$ and $d\ne0$; units or categories in corresponding positions must match.
Quick example
$3/5=x/20$ gives $5x=60$, so $x=12$.
Unit rate
$$\text{unit rate}=\frac{\text{quantity}}{\text{number of units}}$$
Express a comparison per one unit.
Symbols, conditions, and example ＋
What the symbols mean
The numerator is the measured quantity and the denominator counts its units.
Use this formula when
The denominator must be nonzero and both quantities should use compatible units.
Quick example
$180$ miles in $3$ hours is $180/3=60$ miles per hour.
Arithmetic average
$$\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}$$
Share a total equally among all observations.
Symbols, conditions, and example ＋
What the symbols mean
$x_i$ are observations and $n$ is their count.
Use this formula when
$n>0$; the observations should represent quantities that can sensibly be averaged.
Quick example
The average of $6,8,10$ is $24/3=8$.
Percent error
$$\%\text{ error}=\frac{|\text{measured}-\text{accepted}|}{|\text{accepted}|}\cdot100\%$$
Compare measurement error with an accepted reference value.
Symbols, conditions, and example ＋
What the symbols mean
measured is the experimental value and accepted is the reference value.
Use this formula when
The accepted value must be nonzero and both values must use the same units.
Quick example
Measured $9.8$ versus accepted $10$ gives $|9.8-10|/10=2\%$.
Continuous compounding
$$A=Pe^{rt}$$
Model growth when interest is compounded continuously.
Symbols, conditions, and example ＋
What the symbols mean
$P$ is principal, $r$ the annual decimal rate, $t$ time in years, and $A$ the balance.
Use this formula when
The rate and time units must agree; this is an ideal continuous-compounding model.
Quick example
$1000$ at $5\%$ for $2$ years gives $1000e^{0.1}\approx1105.17$.
Zero and negative exponents
$$a^0=1,\qquad a^{-n}=\frac1{a^n}$$
Interpret zero and negative integer powers.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is the nonzero base and $n$ is a positive integer.
Use this formula when
$a\ne0$; $0^0$ is not assigned the value $1$ by this rule.
Quick example
$2^{-3}=1/2^3=1/8$.
Product of two binomials
$$(a+b)(c+d)=ac+ad+bc+bd$$
Expand a product by multiplying every term in one binomial by every term in the other.
Symbols, conditions, and example ＋
What the symbols mean
$a,b,c,d$ may be numbers, variables, or expressions.
Use this formula when
Combine like terms only after all four products are written.
Quick example
$(x+2)(x-3)=x^2-x-6$.
Completing the square
$$x^2+bx=\left(x+\frac b2\right)^2-\left(\frac b2\right)^2$$
Rewrite a quadratic expression as a square plus or minus a constant.
Symbols, conditions, and example ＋
What the symbols mean
$b$ is the coefficient of $x$ after the leading coefficient is made $1$.
Use this formula when
If the original leading coefficient is not $1$, factor it from the quadratic and linear terms first.
Quick example
$x^2+6x=(x+3)^2-9$.
Direct variation
$$y=kx$$
Model two quantities whose ratio stays constant.
Symbols, conditions, and example ＋
What the symbols mean
$k=y/x$ is the constant of variation.
Use this formula when
$x\ne0$ when calculating $k$; the graph passes through the origin.
Quick example
If $y=12$ when $x=3$, then $k=4$ and $y=4x$.
Inverse variation
$$y=\frac{k}{x},\qquad xy=k$$
Model two quantities whose product stays constant.
Symbols, conditions, and example ＋
What the symbols mean
$k$ is the constant of variation.
Use this formula when
$x\ne0$; for physical positive quantities, doubling $x$ halves $y$.
Quick example
If $y=6$ when $x=4$, then $k=24$ and $y=24/x$.
Two-variable linear system
$$\begin{aligned}a_1x+b_1y&=c_1\\a_2x+b_2y&=c_2\end{aligned}\quad x=\frac{c_1b_2-b_1c_2}{a_1b_2-b_1a_2},\quad y=\frac{a_1c_2-c_1a_2}{a_1b_2-b_1a_2}$$
Solve two independent linear equations for two unknowns.
Symbols, conditions, and example ＋
What the symbols mean
The $a_i,b_i$ are coefficients and $c_i$ constants.
Use this formula when
The determinant $a_1b_2-b_1a_2$ must be nonzero for one unique solution.
Quick example
For $x+y=5$ and $x-y=1$, the formulas give $x=3,y=2$.
Binomial theorem
$$(x+y)^n=\sum_{k=0}^{n}\binom nkx^{n-k}y^k$$
Expand a nonnegative integer power of a binomial.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is the power and $k$ indexes each term.
Use this formula when
$n$ must be a nonnegative integer for this finite form.
Quick example
$(x+y)^3=x^3+3x^2y+3xy^2+y^3$.
Remainder theorem
$$\text{remainder of }f(x)\div(x-a)=f(a)$$
Find a linear-division remainder without performing long division.
Symbols, conditions, and example ＋
What the symbols mean
$f$ is a polynomial and $a$ is the zero of the divisor.
Use this formula when
The divisor must have the form $x-a$; $f(a)=0$ exactly when $x-a$ is a factor.
Quick example
For $f(x)=x^2+1$ divided by $x-2$, the remainder is $f(2)=5$.
Triangle angle sum
$$A+B+C=180^\circ=\pi\text{ radians}$$
Relate the three interior angles of a Euclidean triangle.
Symbols, conditions, and example ＋
What the symbols mean
$A,B,C$ are the interior angles.
Use this formula when
This is for a triangle in a flat Euclidean plane.
Quick example
If $A=50^\circ$ and $B=60^\circ$, then $C=70^\circ$.
Exterior-angle sum
$$\sum \text{exterior angles}=360^\circ$$
Add one consistently directed exterior angle at every vertex of a polygon.
Symbols, conditions, and example ＋
What the symbols mean
Each exterior angle is the turning angle at a vertex.
Use this formula when
Use one exterior angle per vertex and traverse the polygon in one direction.
Quick example
A regular octagon has each exterior angle $360^\circ/8=45^\circ$.
Regular polygon angles
$$\text{interior angle}=\frac{(n-2)180^\circ}{n},\qquad \text{central angle}=\frac{360^\circ}{n}$$
Find equal interior and central angles of a regular polygon.
Symbols, conditions, and example ＋
What the symbols mean
$n$ is the number of sides.
Use this formula when
$n\ge3$ and the polygon must be regular.
Quick example
A regular hexagon has interior angles $120^\circ$ and central angles $60^\circ$.
Rectangular prism volume and surface area
$$V=\ell wh,\qquad S=2(\ell w+\ell h+wh)$$
Measure the space inside and total exterior area of a rectangular prism.
Symbols, conditions, and example ＋
What the symbols mean
$\ell,w,h$ are length, width, and perpendicular height.
Use this formula when
All dimensions must be nonnegative and use the same unit.
Quick example
For $\ell=3,w=4,h=5$, $V=60$ and $S=94$.
Cube volume and surface area
$$V=s^3,\qquad S=6s^2$$
Measure a cube using its common edge length.
Symbols, conditions, and example ＋
What the symbols mean
$s$ is the length of every edge.
Use this formula when
$s\ge0$; volume uses cubic units and area square units.
Quick example
For $s=4$, $V=64$ and $S=96$.
Prism surface area
$$S=2B+Ph$$
Find the total area of two congruent bases and all lateral faces of a right prism.
Symbols, conditions, and example ＋
What the symbols mean
$B$ is base area, $P$ base perimeter, and $h$ prism height.
Use this formula when
The prism must be right for the lateral area $Ph$; otherwise use slant edge geometry.
Quick example
If $B=12,P=14,h=5$, then $S=24+70=94$.
Similarity scale factors
$$\frac{L_2}{L_1}=k,\qquad\frac{A_2}{A_1}=k^2,\qquad\frac{V_2}{V_1}=k^3$$
Relate corresponding lengths, areas, and volumes of similar figures.
Symbols, conditions, and example ＋
What the symbols mean
$k$ is the linear scale factor.
Use this formula when
The figures or solids must be similar and corresponding measurements must be matched.
Quick example
If lengths double, areas multiply by $4$ and volumes by $8$.
Distance in three dimensions
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$
Find straight-line distance between two points in three-dimensional Cartesian space.
Symbols, conditions, and example ＋
What the symbols mean
$(x_i,y_i,z_i)$ are endpoint coordinates.
Use this formula when
All coordinate axes must use the same length scale.
Quick example
Between $(0,0,0)$ and $(1,2,2)$, $d=\sqrt9=3$.
Equilateral triangle measures
$$h=\frac{\sqrt3}{2}s,\qquad A=\frac{\sqrt3}{4}s^2$$
Find the height and area of an equilateral triangle from one side.
Symbols, conditions, and example ＋
What the symbols mean
$s$ is the common side length.
Use this formula when
$s\ge0$ and all three sides must be equal.
Quick example
For $s=6$, $h=3\sqrt3$ and $A=9\sqrt3$.
Chord length
$$c=2r\sin\left(\frac{\theta}{2}\right)$$
Find the straight chord subtended by a central angle.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radius and $\theta$ the smaller central angle.
Use this formula when
Use a consistent angle mode; the sine form works directly in radians or degrees when the calculator mode matches.
Quick example
For $r=5$ and $\theta=60^\circ$, $c=10\sin30^\circ=5$.
Quotient identities
$$\tan\theta=\frac{\sin\theta}{\cos\theta},\qquad\cot\theta=\frac{\cos\theta}{\sin\theta}$$
Express tangent and cotangent using sine and cosine.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is an angle where the denominator is nonzero.
Use this formula when
For tangent, $\cos\theta\ne0$; for cotangent, $\sin\theta\ne0$.
Quick example
At $45^\circ$, $\tan\theta=(\sqrt2/2)/(\sqrt2/2)=1$.
Cofunction identities
$$\sin\left(\frac\pi2-\theta\right)=\cos\theta,\qquad\cos\left(\frac\pi2-\theta\right)=\sin\theta$$
Connect complementary angles and paired trigonometric functions.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is any angle for which the functions are defined.
Use this formula when
In degree mode replace $\pi/2$ with $90^\circ$.
Quick example
$\sin30^\circ=\cos60^\circ=1/2$.
Unit-circle coordinates
$$(x,y)=(\cos\theta,\sin\theta),\qquad\tan\theta=\frac yx$$
Read trigonometric values from a point on the unit circle.
Symbols, conditions, and example ＋
What the symbols mean
$\theta$ is measured from the positive $x$-axis.
Use this formula when
The point must satisfy $x^2+y^2=1$; tangent requires $x\ne0$.
Quick example
At $\theta=\pi/6$, the point is $(\sqrt3/2,1/2)$.
Sinusoid amplitude and period
$$y=A\sin(B(x-C))+D,\quad \text{amplitude}=|A|,\quad \text{period}=\frac{2\pi}{|B|}$$
Read the shape and transformations of a sinusoidal model.
Symbols, conditions, and example ＋
What the symbols mean
$C$ is horizontal shift and $D$ the midline.
Use this formula when
$B\ne0$; for degrees use period $360^\circ/|B|$.
Quick example
$y=3\sin(2x)+1$ has amplitude $3$, period $\pi$, and midline $y=1$.
Product-to-sum identities
$$2\sin A\cos B=\sin(A+B)+\sin(A-B)$$
Rewrite a product of trigonometric functions as a sum.
Symbols, conditions, and example ＋
What the symbols mean
$A,B$ are angle expressions.
Use this formula when
Use matching angle units throughout; other product-to-sum forms have different signs.
Quick example
$2\sin3x\cos x=\sin4x+\sin2x$.
Complex polar form
$$z=r(\cos\theta+i\sin\theta)=re^{i\theta},\qquad r=\sqrt{x^2+y^2}$$
Represent a complex number by magnitude and direction.
Symbols, conditions, and example ＋
What the symbols mean
$z=x+iy$, $r$ is modulus, and $\theta$ an argument.
Use this formula when
For $z\ne0$, arguments differ by multiples of $2\pi$; the zero argument is undefined.
Quick example
$1+i=\sqrt2e^{i\pi/4}$.
De Moivre’s theorem
$$[r(\cos\theta+i\sin\theta)]^n=r^n[\cos(n\theta)+i\sin(n\theta)]$$
Raise a complex number in polar form to an integer power.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is modulus, $\theta$ an argument, and $n$ an integer.
Use this formula when
Use radians consistently when calculating the angle numerically.
Quick example
$(\cos\frac\pi3+i\sin\frac\pi3)^3=\cos\pi+i\sin\pi=-1$.
Function composition
$$(f\circ g)(x)=f(g(x))$$
Use the output of one function as the input of another.
Symbols, conditions, and example ＋
What the symbols mean
$g$ is applied first and $f$ second.
Use this formula when
$x$ must lie in the domain of $g$ and $g(x)$ in the domain of $f$.
Quick example
If $f(x)=x^2$ and $g(x)=x+1$, then $(f\circ g)(x)=(x+1)^2$.
Inverse function relationship
$$f^{-1}(f(x))=x,\qquad f(f^{-1}(x))=x$$
Undo a one-to-one function with its inverse.
Symbols, conditions, and example ＋
What the symbols mean
$f^{-1}$ denotes inverse function, not reciprocal.
Use this formula when
Restrict the domain when needed so $f$ is one-to-one; inputs must remain in the appropriate domains.
Quick example
For $f(x)=2x+3$, $f^{-1}(x)=(x-3)/2$.
Function transformation
$$y=a\,f(b(x-h))+k$$
Describe vertical scale, horizontal scale, reflection, and translation of a graph.
Symbols, conditions, and example ＋
What the symbols mean
$h,k$ shift the graph and $a,b$ scale or reflect it.
Use this formula when
$a\ne0$ and $b\ne0$ for a noncollapsed transformed graph; horizontal scale is $1/|b|$.
Quick example
$2f(3(x-1))-4$ shifts right $1$, scales horizontally by $1/3$, vertically by $2$, and down $4$.
Remaining trigonometric derivatives
$$(\cot x)'=-\csc^2x,\quad(\sec x)'=\sec x\tan x,\quad(\csc x)'=-\csc x\cot x$$
Differentiate cotangent, secant, and cosecant.
Symbols, conditions, and example ＋
What the symbols mean
$x$ is measured in radians.
Use this formula when
Each function and the expression on the right must be defined at the point.
Quick example
$\frac{d}{dx}\sec(2x)=2\sec(2x)\tan(2x)$.
Remaining inverse-trig derivatives
$$(\arccos x)'=-\frac1{\sqrt{1-x^2}},\quad(\operatorname{arccot}x)'=-\frac1{1+x^2}$$
Differentiate inverse cosine and the common decreasing convention for inverse cotangent.
Symbols, conditions, and example ＋
What the symbols mean
$x$ is the real input.
Use this formula when
For arccos, $|x|<1$; inverse-cotangent conventions can vary by textbook.
Quick example
$\frac{d}{dx}\arccos(3x)=-3/\sqrt{1-9x^2}$.
Common exponential and trigonometric antiderivatives
$$\int e^x dx=e^x+C,\quad\int\cos x\,dx=\sin x+C,\quad\int\sin x\,dx=-\cos x+C$$
Recall three fundamental antiderivatives.
Symbols, conditions, and example ＋
What the symbols mean
$C$ is an arbitrary constant.
Use this formula when
Angles are in radians; include an inner-derivative adjustment for composite inputs.
Quick example
$\int\cos(3x)dx=\frac13\sin(3x)+C$.
L’Hôpital’s rule
$$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$$
Evaluate certain indeterminate quotients by differentiating numerator and denominator separately.
Symbols, conditions, and example ＋
What the symbols mean
$f,g$ are differentiable near the limit point.
Use this formula when
The original quotient must have form $0/0$ or $\infty/\infty$, and the derivative quotient limit must satisfy the theorem.
Quick example
$\lim_{x\to0}\sin x/x=\lim_{x\to0}\cos x/1=1$.
Mean value theorem
$$f'(c)=\frac{f(b)-f(a)}{b-a}\quad\text{for some }c\in(a,b)$$
Guarantee a tangent slope equal to the interval’s average rate of change.
Symbols, conditions, and example ＋
What the symbols mean
$a,b$ are endpoints and $c$ is an interior point.
Use this formula when
$f$ must be continuous on $[a,b]$ and differentiable on $(a,b)$.
Quick example
For $f(x)=x^2$ on $[1,3]$, average slope is $4$, so $c=2$.
Position, velocity, and acceleration
$$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t)$$
Relate an object’s position to instantaneous velocity and acceleration.
Symbols, conditions, and example ＋
What the symbols mean
$s$ is position, $v$ velocity, $a$ acceleration, and $t$ time.
Use this formula when
Keep units consistent; speed is $|v|$ and total distance may require splitting where $v$ changes sign.
Quick example
If $s=t^3$, then $v=3t^2$ and $a=6t$.
Parametric arc length
$$L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$$
Measure length along a parametrized plane curve.
Symbols, conditions, and example ＋
What the symbols mean
$t$ is the parameter and $[a,b]$ its interval.
Use this formula when
The derivatives should be continuous and the integral must converge.
Quick example
For $x=3t,y=4t$ on $[0,2]$, $L=\int_0^2 5dt=10$.
Surface area of revolution
$$S=2\pi\int_a^b r(x)\sqrt{1+[f'(x)]^2}\,dx$$
Add the curved bands formed by revolving a graph around an axis.
Symbols, conditions, and example ＋
What the symbols mean
$r(x)$ is nonnegative distance from the curve to the rotation axis.
Use this formula when
The curve must be sufficiently smooth; choose radius and variable to match the axis and slicing direction.
Quick example
About the $x$-axis for $y=f(x)\ge0$, use $r(x)=f(x)$.
Polar arc length
$$L=\int_{\alpha}^{\beta}\sqrt{r(\theta)^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$$
Measure length along a polar curve.
Symbols, conditions, and example ＋
What the symbols mean
$r$ is radial distance and $\theta$ the polar angle.
Use this formula when
$r$ and $dr/d\theta$ should be continuous on the interval and angles use radians.
Quick example
For $r=2$, $0\le\theta\le\pi$, $L=\int_0^\pi2d\theta=2\pi$.
Common Maclaurin series
$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!},\quad\sin x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!},\quad\cos x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}$$
Use standard power series for exponential, sine, and cosine.
Symbols, conditions, and example ＋
What the symbols mean
$n$ indexes terms and factorials determine coefficients.
Use this formula when
These three series converge for every real or complex $x$.
Quick example
Near $0$, $\sin x\approx x-x^3/6$.
Geometric series test
$$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}\quad\text{when }|r|<1$$
Determine convergence and sum of an infinite geometric series.
Symbols, conditions, and example ＋
What the symbols mean
$a$ is the first term and $r$ the common ratio.
Use this formula when
The series diverges when $|r|\ge1$.
Quick example
$1+1/2+1/4+\cdots=1/(1-1/2)=2$.
Ratio test
$$L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$$
Test absolute convergence by comparing successive term sizes.
Symbols, conditions, and example ＋
What the symbols mean
$a_n$ is the series term and $L$ the limiting ratio.
Use this formula when
The series converges if $L<1$, diverges if $L>1$, and the test is inconclusive if $L=1$.
Quick example
For $\sum1/n!$, the ratio is $1/(n+1)\to0$, so it converges.
Alternating-series error bound
$$|R_n|\le b_{n+1}$$
Bound the error after stopping a convergent alternating series.
Symbols, conditions, and example ＋
What the symbols mean
$b_n$ is the positive magnitude of the $n$th alternating term.
Use this formula when
The magnitudes must decrease and approach zero.
Quick example
For $1-1/2+1/3-\cdots$, the error after $n$ terms is at most $1/(n+1)$.
Improper integral definition
$$\int_a^{\infty}f(x)dx=\lim_{b\to\infty}\int_a^b f(x)dx$$
Define an integral over an unbounded interval using a limit.
Symbols, conditions, and example ＋
What the symbols mean
$b$ is a temporary finite endpoint.
Use this formula when
The improper integral converges only when the displayed limit exists and is finite.
Quick example
$\int_1^{\infty}x^{-2}dx=\lim_{b\to\infty}[-1/x]_1^b=1$.
Range and interquartile range
$$\text{range}=x_{\max}-x_{\min},\qquad IQR=Q_3-Q_1$$
Measure total spread and the spread of the middle half of ordered data.
Symbols, conditions, and example ＋
What the symbols mean
$Q_1,Q_3$ are the first and third quartiles.
Use this formula when
Sort the data first; quartile conventions can vary slightly for small samples.
Quick example
For quartiles $Q_1=4,Q_3=11$, $IQR=7$.
Normal density
$$f(x)=\frac1{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}$$
Describe the bell-shaped normal probability density.
Symbols, conditions, and example ＋
What the symbols mean
$\mu$ is mean and $\sigma$ standard deviation.
Use this formula when
$\sigma>0$; probabilities are areas under the density, not density heights alone.
Quick example
The standard normal uses $\mu=0$ and $\sigma=1$.
Margin of error
$$ME=(\text{critical value})(\text{standard error})$$
Express the half-width of a confidence interval.
Symbols, conditions, and example ＋
What the symbols mean
The critical value comes from a confidence level and reference distribution.
Use this formula when
Use a standard-error formula and critical distribution appropriate to the parameter and assumptions.
Quick example
If $z^*=1.96$ and $SE=0.03$, then $ME=0.0588$.
Two-sample t statistic
$$t=\frac{(\bar x_1-\bar x_2)-\Delta_0}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$$
Compare two independent population means without assuming known population spreads.
Symbols, conditions, and example ＋
What the symbols mean
$\Delta_0$ is the null difference, usually zero.
Use this formula when
Samples should be independent and conditions for t inference should be checked; use Welch degrees of freedom unless pooling is justified.
Quick example
Substitute both sample means, standard deviations, and sizes before comparing with a t distribution.
Two-proportion z statistic
$$z=\frac{(\hat p_1-\hat p_2)-0}{\sqrt{\hat p(1-\hat p)(1/n_1+1/n_2)}}$$
Test whether two independent population proportions differ.
Symbols, conditions, and example ＋
What the symbols mean
$\hat p$ is the pooled success proportion under the equal-proportions null.
Use this formula when
Samples must be independent and expected success and failure counts sufficiently large.
Quick example
Pool successes across both groups only for the null-test standard error.
Coefficient of determination
$$R^2=1-\frac{SS_{\mathrm{res}}}{SS_{\mathrm{tot}}}$$
Measure the proportion of response variation explained by a fitted model.
Symbols, conditions, and example ＋
What the symbols mean
$SS_{\mathrm{res}}$ is residual sum of squares and $SS_{\mathrm{tot}}$ total sum of squares.
Use this formula when
Interpret within the fitted data and model; a large $R^2$ does not prove causation or a correct model.
Quick example
If residual variation is $20$ of total variation $100$, then $R^2=0.80$.
Law of total probability
$$P(B)=\sum_i P(B\mid A_i)P(A_i)$$
Combine conditional probabilities across every mutually exclusive case.
Symbols, conditions, and example ＋
What the symbols mean
The events $A_i$ form a partition of the sample space.
Use this formula when
The $A_i$ must be disjoint, exhaustive, and have positive probability when conditioned upon.
Quick example
If two groups have shares $.4,.6$ and event rates $.2,.5$, then $P(B)=.4(.2)+.6(.5)=.38$.
Independence test
$$A\perp B\iff P(A\cap B)=P(A)P(B)$$
Check whether knowing one event changes the probability of another.
Symbols, conditions, and example ＋
What the symbols mean
$A,B$ are events.
Use this formula when
Do not confuse independence with mutually exclusive events; nonempty disjoint events are generally dependent.
Quick example
If $P(A)=.5,P(B)=.4,P(A\cap B)=.2$, then $A$ and $B$ are independent.
Uniform distribution
$$X\sim U(a,b),\quad f(x)=\frac1{b-a},\quad E[X]=\frac{a+b}{2},\quad\operatorname{Var}(X)=\frac{(b-a)^2}{12}$$
Model an outcome equally likely across a finite interval.
Symbols, conditions, and example ＋
What the symbols mean
$a$ and $b$ are lower and upper endpoints.
Use this formula when
$a<b$ and the density is zero outside $[a,b]$.
Quick example
For $U(0,10)$, $P(2<X<5)=3/10$.
Exponential distribution
$$f(x)=\lambda e^{-\lambda x},\quad E[X]=\frac1\lambda,\quad P(X>x)=e^{-\lambda x}$$
Model waiting time between constant-rate Poisson events.
Symbols, conditions, and example ＋
What the symbols mean
$\lambda$ is the positive event rate.
Use this formula when
$x\ge0$, $\lambda>0$, and the model has the memoryless property.
Quick example
For $\lambda=2$, $P(X>1)=e^{-2}$.
Hypergeometric probability
$$P(X=k)=\frac{\binom Kk\binom{N-K}{n-k}}{\binom Nn}$$
Find successes when sampling without replacement from a finite population.
Symbols, conditions, and example ＋
What the symbols mean
$N$ is population size, $K$ successes, $n$ draws, and $k$ observed successes.
Use this formula when
Selections are without replacement and all size-$n$ samples are equally likely.
Quick example
From $N=10,K=4,n=3$, exactly $k=2$ has probability $\binom42\binom61/\binom{10}3$.
Matrix transpose
$$(A^T)_{ij}=A_{ji}$$
Turn matrix rows into columns.
Symbols, conditions, and example ＋
What the symbols mean
$i,j$ index row and column positions.
Use this formula when
For products, $(AB)^T=B^TA^T$; transposing twice returns the original matrix.
Quick example
$\begin{bmatrix}1&2\\3&4\end{bmatrix}^T=\begin{bmatrix}1&3\\2&4\end{bmatrix}$.
Trace
$$\operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii}$$
Add the main-diagonal entries of a square matrix.
Symbols, conditions, and example ＋
What the symbols mean
$a_{ii}$ are diagonal entries.
Use this formula when
$A$ must be square; trace is unchanged by a cyclic reordering such as $\operatorname{tr}(AB)=\operatorname{tr}(BA)$.
Quick example
For diagonal entries $2,3,5$, the trace is $10$.
Cramer’s rule
$$A\mathbf x=\mathbf b,\qquad x_i=\frac{\det(A_i)}{\det(A)}$$
Express each coordinate of a unique square-system solution using determinants.
Symbols, conditions, and example ＋
What the symbols mean
$A_i$ replaces column $i$ of $A$ with $\mathbf b$.
Use this formula when
$A$ must be square and $\det(A)\ne0$; elimination is usually faster for large systems.
Quick example
For a two-variable system, replace one coefficient column at a time and divide its determinant by $\det(A)$.
Homogeneous solution forms
$$r_1\ne r_2:\ y=C_1e^{r_1x}+C_2e^{r_2x};\quad r:\ y=(C_1+C_2x)e^{rx};\quad \alpha\pm i\beta:\ y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)$$
Write the real general solution from characteristic roots.
Symbols, conditions, and example ＋
What the symbols mean
$r_1,r_2$ are roots and $C_1,C_2$ arbitrary constants.
Use this formula when
This applies to second-order homogeneous linear equations with constant coefficients.
Quick example
Roots $\pm i$ give $y=C_1\cos x+C_2\sin x$.
Fourth-order Runge–Kutta
$$y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4)$$
Advance an initial-value solution using four slope samples.
Symbols, conditions, and example ＋
What the symbols mean
$k_1=f(x_n,y_n)$, $k_2=f(x_n+h/2,y_n+hk_1/2)$, $k_3=f(x_n+h/2,y_n+hk_2/2)$, and $k_4=f(x_n+h,y_n+hk_3)$.
Use this formula when
Use a sufficiently smooth differential equation and monitor numerical error as step size changes.
Quick example
RK4 is generally much more accurate per step than Euler’s method for smooth problems.
Pigeonhole principle
$$N\text{ objects in }k\text{ boxes}\implies\text{some box has at least }\left\lceil\frac Nk\right\rceil\text{ objects}$$
Guarantee a repeated placement without identifying which box receives it.
Symbols, conditions, and example ＋
What the symbols mean
$N$ is the object count and $k$ the nonempty collection of boxes.
Use this formula when
$N\ge0$ and $k>0$ are integers.
Quick example
Among $13$ people, at least $\lceil13/12\rceil=2$ share a birth month.
Edges in a complete graph
$$|E(K_n)|=\binom n2=\frac{n(n-1)}2$$
Count edges joining every pair of distinct vertices.
Symbols, conditions, and example ＋
What the symbols mean
$K_n$ is the simple complete graph on $n$ vertices.
Use this formula when
No loops or parallel edges are included.
Quick example
$K_5$ has $5(4)/2=10$ edges.
Edges in a tree
$$|E|=|V|-1$$
Relate edge and vertex counts in a finite tree.
Symbols, conditions, and example ＋
What the symbols mean
$V$ is the vertex set and $E$ the edge set.
Use this formula when
The graph must be connected and contain no cycles.
Quick example
A tree with $12$ vertices has $11$ edges.
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Coverage across the mathematics pathway
This collection covers the central computational formulas students repeatedly use across the site’s courses. Specialized courses can require additional definitions and theorems, so each subject has its own page grouping every formula by topic, alongside a fuller subject guide and calculator.
Arithmetic
Every arithmetic formula grouped by topic, with its restrictions and worked substitutions.
Algebra
Every algebra formula grouped by topic, with its restrictions and worked substitutions.
Geometry
Every geometry formula grouped by topic, with its restrictions and worked substitutions.
Trigonometry
Every trigonometry formula grouped by topic, with its restrictions and worked substitutions.
Precalculus
Every precalculus formula grouped by topic, with its restrictions and worked substitutions.
Calculus
Every calculus formula grouped by topic, with its restrictions and worked substitutions.
Statistics
Every statistics formula grouped by topic, with its restrictions and worked substitutions.
Probability
Every probability formula grouped by topic, with its restrictions and worked substitutions.
Linear Algebra
Every linear algebra formula grouped by topic, with its restrictions and worked substitutions.
Differential Equations
Every differential equations formula grouped by topic, with its restrictions and worked substitutions.
Discrete Math
Every discrete math formula grouped by topic, with its restrictions and worked substitutions.
Questions about using math formulas
How do I know which formula to use?
Start from what the question asks for and what it gives you. Match those quantities to the symbols in a formula. If a formula contains a quantity you were not given and cannot work out, it is the wrong formula for that question.
Why does a formula have conditions?
A formula is only true when its assumptions hold. The Pythagorean theorem needs a right angle, and dividing by a variable needs that variable to be non-zero. Checking the condition first is what stops a correct calculation from giving a wrong answer.
Should I memorise formulas or look them up?
Memorise the handful you use every week, and look up the rest. What matters more is knowing what each symbol means and when the formula applies, because that is what tells you whether your answer is sensible.
How do I check an answer I got from a formula?
Put the answer back into the original relationship and see if both sides agree. Then check the size, the sign, and the unit against a rough estimate. An answer that is the right number with the wrong unit is still wrong.