Area of a Triangle

Geometry is the math of shape, size, and position. You use it to find lengths, angles, areas, and volumes.

Area of a Triangle: the central idea

Triangle area is half the product of any base and its corresponding perpendicular height because two congruent copies form a parallelogram.

Words you need

Base

The base is whichever side you choose to measure the height from, and a triangle has three legal choices of base that all give the same area.

Perpendicular height

The perpendicular height, also called the altitude, is the shortest straight-out distance from the base line to the opposite corner, meeting the base at $90^\circ$.

Included angle

The included angle is the angle sitting between the two sides you know, and it is the angle $C$ that the formula $A=\tfrac12ab\sin C$ needs.

Semi-perimeter

The semi-perimeter, written $s$, is half of the triangle's perimeter, so $s=(a+b+c)/2$, and it is the first quantity Heron's formula asks for.

Heron's formula

Heron's formula computes area from the three side lengths alone using $A=\sqrt{s(s-a)(s-b)(s-c)}$, with no angle and no height required.

Square units

Square units are the units area is reported in, such as square centimetres or square feet, because area multiplies one length by another length.

What to know before this lesson

Know perpendicular distance, multiplication of fractions, and how to choose a base without changing the triangle.

If one of those prerequisites is uncertain, use the Geometry subject guide to locate the earlier concept before memorizing a procedure.

The area of a triangle: a worked example

Every step, with the arithmetic

1. Step 1 - See what you haveThe sides are $a=13$, $b=14$, $c=15$. No height is given, so use Heron's formula.
2. Step 2 - Add the three sides$13+14+15=42$
3. Step 3 - Halve that to get the semi-perimeter$s=42\div2=21$
4. Step 4 - Subtract each side from $s$$21-13=8$, $21-14=7$, $21-15=6$
5. Step 5 - Multiply the four numbers$21\times8\times7\times6=7056$
6. Step 6 - Take the square root$A=\sqrt{7056}=84$ square units
7. Step 7 - Cross-check with base and heightUsing $14$ as the base, $14h/2=84$, so $7h=84$ and $h=12$. A $13$-$14$-$15$ triangle really does stand $12$ units tall over its $14$-unit side.

A base of $9$ and perpendicular height $4$ produces $A=(9)(4)/2=18$ square units.

Pick the area formula that matches the information you were given

The step-by-step method for the area of a triangle

1. Choose one side as the base and identify the perpendicular distance from that base line to the opposite vertex.
2. Convert base and height to the same length unit.
3. Compute $A=bh/2$ and report square units; use Heron's or a trigonometric form when height is unavailable.

How to check your answer

Double the triangle area and compare it with the $9\times4=36$ area of the related parallelogram.

Test triangle inequalities, angle sums, units, scale, and whether the result is compatible with the diagram without assuming the drawing is exact.

A mistake that changes the mathematics

A slanted side is not the height unless it is perpendicular to the chosen base.

Where you will actually use this

Painting a gable end

The triangular wall above a house's top floor is $24$ feet wide at its base and rises $7$ feet to the roof peak. Its area is $24\times7\div2=84$ square feet, so one gallon of paint covering $350$ square feet is plenty for two coats.

Sail making

A triangular sail with a $4$-metre foot and a perpendicular height of $9$ metres needs $4\times9\div2=18$ square metres of cloth, before seams and hems are added.

Sizing a corner land plot

A surveyor measures two boundaries of $120$ feet and $90$ feet meeting at $40^\circ$. The plot's area is $\tfrac12(120)(90)\sin40^\circ\approx5400\times0.643\approx3471$ square feet, which is what the tax assessment is based on.

How the area of a triangle connects to the rest of geometry

• Pythagorean theorem — When only the sides of an isosceles triangle are known, the theorem finds the missing perpendicular height first, and then $A=bh/2$ finishes the job.
• Sine and cosine rules — The formula $A=\tfrac12ab\sin C$ uses the same side and angle labels as the sine and cosine rules, so the two topics solve triangles as one toolkit.
• Geometric transformations — Sliding the top corner sideways along a line parallel to the base keeps both the base and the height the same, so the area never changes; a dilation by scale factor $k$ multiplies area by $k^2$.

Try a transfer problem

Find the area of a triangle with sides $5$ and $8$ enclosing a $30^\circ$ angle using $A=\tfrac12ab\sin C$.

Show the worked answer

Use $A=\tfrac12ab\sin C$ with $a=5$, $b=8$, and the included angle $C=30^\circ$. Since $\sin30^\circ=0.5$ exactly, $A=\tfrac12(5)(8)(0.5)=\tfrac12(40)(0.5)=20\times0.5=10$ square units. A quick sanity check: the tallest this triangle could ever be over its $8$-unit base is $5$, which would give an area of $20$. Because the $30^\circ$ angle makes the real height only $5\sin30^\circ=2.5$, an answer of $10$ is exactly half that ceiling, which fits.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to geometry practice questions.

Questions about the area of a triangle

Can I use any side as the base?

Yes. Every triangle has three base-and-height pairs, and all three give the same area. In the $13$-$14$-$15$ triangle, the $14$ side pairs with height $12$ and the $15$ side pairs with height $11.2$; both give $84$ square units.

Why do you divide by two?

Because two identical copies of a triangle snap together into a parallelogram with the same base and height. That parallelogram has area $bh$, and your triangle is exactly half of it.

What if the height lands outside the triangle?

That happens with obtuse triangles, and the formula still works. Extend the base line beyond the triangle until the perpendicular from the far corner reaches it, and measure the height there. The area is still $bh/2$.

How do I find area from three coordinate points?

Use the shoelace formula $A=\tfrac12|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$. For $(0,0)$, $(6,0)$, and $(0,4)$ this gives $\tfrac12|0(0-4)+6(4-0)+0(0-0)|=\tfrac12(24)=12$ square units. The absolute value bars keep the answer positive no matter what order you list the points in.