Pythagorean Theorem

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

Pythagorean Theorem: the central idea

The Pythagorean theorem relates only the side lengths of a right triangle: the sum of the squared legs equals the squared hypotenuse.

Words you need

Right angle

A right angle is a square corner that measures exactly $90^\circ$, and it is the one angle the Pythagorean theorem requires a triangle to have.

Hypotenuse

The hypotenuse is the side that sits directly across from the right angle, and it is always the longest side of a right triangle.

Leg

A leg is one of the two shorter sides that meet to form the right angle, and the two legs are the sides squared and added on the left of $a^2+b^2=c^2$.

Converse

The converse of the Pythagorean theorem is the rule read backwards: it says that if three side lengths satisfy $a^2+b^2=c^2$, then the triangle must contain a right angle.

Pythagorean triple

A Pythagorean triple is a set of three whole numbers, such as $3$-$4$-$5$ or $5$-$12$-$13$, that satisfies $a^2+b^2=c^2$ exactly with no decimals left over.

Distance formula

The distance formula, $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, is the Pythagorean theorem written for a grid, where the sideways gap and the up-and-down gap act as the two legs.

What to know before this lesson

Recognize right angles, square numbers, square roots, and the names leg and hypotenuse.

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

The Pythagorean theorem: a worked example

Every step, with the arithmetic

1. Step 1 - Name the sidesThe hypotenuse is $c=20$ because it lies across from the right angle. One leg is $a=12$. The other leg is $b$, the unknown.
2. Step 2 - Write the theorem with the numbers in it$12^2+b^2=20^2$
3. Step 3 - Square the two known lengths$144+b^2=400$
4. Step 4 - Get $b^2$ by itself$b^2=400-144=256$
5. Step 5 - Take the positive square root$b=\sqrt{256}=16$. A side length is a distance, so the negative root is thrown out.
6. Step 6 - Check the answer in the original rule$12^2+16^2=144+256=400$, and $20^2=400$. They match.
7. Step 7 - Notice the pattern$12$-$16$-$20$ is every side of $3$-$4$-$5$ multiplied by $4$. Scaling a right triangle keeps it a right triangle.

With legs $6$ and $8$, $c^2=36+64=100$. A geometric length is positive, so $c=10$ rather than $\pm10$.

Common Pythagorean triples

The step-by-step method for the Pythagorean theorem

1. Confirm the triangle is right and label the side opposite the right angle as $c$.
2. Substitute the known side lengths into $a^2+b^2=c^2$ without assigning the hypotenuse as a leg.
3. Solve for the missing positive length and test whether it satisfies the triangle inequality.

How to check your answer

Square the three lengths: $36+64=100$. The equality confirms the computed side and can also classify a $6$-$8$-$10$ triangle as right.

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

The longest visible side is not automatically the hypotenuse; it earns that name only when it lies opposite a proven right angle.

Where you will actually use this

TV and monitor screen sizes

A screen is sold by its diagonal, which is the hypotenuse of the picture. A panel $40$ inches wide and $30$ inches tall has a diagonal of $\sqrt{1600+900}=\sqrt{2500}=50$ inches, so it is sold as a $50$-inch TV.

Squaring up a building

Builders check corners with the $3$-$4$-$5$ method. Measure $3$ feet along one wall and $4$ feet along the other; if the gap between the two marks is exactly $5$ feet, the corner is a true right angle. Off by a little and the wall is out of square.

Straight-line distance on a map

Walk $300$ metres east and then $400$ metres north and you are $\sqrt{300^2+400^2}=\sqrt{250000}=500$ metres from where you started. GPS software uses this same distance formula on coordinates.

How the Pythagorean theorem connects to the rest of geometry

• Distance formula and circles — A circle is every point one radius away from the center, and that distance is measured with the Pythagorean theorem, which is why the circle equation is full of squares.
• The unit circle — Each point on the unit circle sits on a right triangle with legs $\cos\theta$ and $\sin\theta$ and hypotenuse $1$, so the theorem turns straight into $\cos^2\theta+\sin^2\theta=1$.
• The cosine rule — The cosine rule $c^2=a^2+b^2-2ab\cos C$ handles triangles with no square corner; set $C=90^\circ$ and the last piece disappears, leaving $a^2+b^2=c^2$.

Try a transfer problem

A ladder is $13$ feet long and stands $5$ feet from a wall. Find its height and draw the right angle that justifies the theorem.

Show the worked answer

The ladder is the hypotenuse, so $c=13$. The gap along the ground is one leg, $a=5$, and the height up the wall is the other leg, $b$. The right angle is where the wall meets the ground. Substituting gives $5^2+b^2=13^2$, so $25+b^2=169$ and $b^2=169-25=144$. Taking the positive square root gives $b=\sqrt{144}=12$ feet. The ladder reaches $12$ feet up the wall. Notice the sides are $5$-$12$-$13$, a Pythagorean triple, which is why the answer came out as a whole number.

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 Pythagorean theorem

Is 3-4-5 always a right triangle?

Yes. Because $3^2+4^2=9+16=25=5^2$, the converse of the Pythagorean theorem guarantees a right angle. The same holds for every multiple, such as $6$-$8$-$10$ or $30$-$40$-$50$, because scaling all three sides by the same number keeps the angles unchanged.

How do I tell which side is the hypotenuse?

Find the right angle first, then follow your eye straight across the triangle. The side you land on is the hypotenuse. It is also the longest side, so if you plug the longest number in as a leg, your answer will come out too small or the arithmetic will give a negative square.

Does the theorem work on triangles without a right angle?

No. For a triangle with no square corner, use the cosine rule $c^2=a^2+b^2-2ab\cos C$ instead. Comparing $a^2+b^2$ with $c^2$ still tells you something useful: if the sum is bigger the triangle is acute, and if it is smaller the triangle is obtuse.

What are the 45-45-90 and 30-60-90 side ratios?

Both come straight from the theorem. A $45$-$45$-$90$ triangle has sides in the ratio $1:1:\sqrt2$, since $1^2+1^2=2$. A $30$-$60$-$90$ triangle has sides in the ratio $1:\sqrt3:2$, since $1^2+(\sqrt3)^2=1+3=4=2^2$. Memorizing these two saves you the arithmetic on most test questions.