The Pythagorean Theorem, Step by Step

In a right triangle (one with a 90° corner), the square of the longest side equals the sum of the squares of the other two: a² + b² = c². The longest side, c, is called the hypotenuse. It sits opposite the right angle.

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The steps

Identify the hypotenuse c (opposite the right angle). The other two sides are the legs a and b.
To find the hypotenuse: square both legs, add them, and take the square root.
To find a missing leg: square both known sides, subtract the leg squared from the hypotenuse squared, and take the square root.
Check that the hypotenuse is the longest side.
Include the units.

A worked example

A right triangle has legs 9 and 12. Find the hypotenuse.

a = 9 and b = 12.
Square them: 9² = 81 and 12² = 144.
Add: 81 + 144 = 225, so c² = 225.
Take the square root: c = √225 = 15.

The hypotenuse is 15. Check: 15 is longer than both legs, and 9-12-15 is 3 times the 3-4-5 triangle. ✓

Why it works

Here is a picture. Draw a square on each side of a right triangle. The big square on the hypotenuse has exactly the same area as the two smaller squares added together. With legs 3 and 4: 9 + 16 = 25, which is 5².

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Work it out: hypotenuse 9 12Opens The Shape Shop with this example

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Common mistakes

Adding when finding a leg. For a missing leg, subtract: a² = c² − b².
Forgetting the square root. 225 is c², not c.
Using it on a triangle without a right angle. The theorem only works for right triangles.
Squaring by doubling. 9² is 81, not 18.

Questions people ask

How do I find a missing leg?

With hypotenuse 13 and one leg 5: 13² − 5² = 169 − 25 = 144, and √144 = 12.

What are Pythagorean triples?

Sets of whole numbers that fit, like 3-4-5, 5-12-13, 8-15-17 and 7-24-25.

How is it used for distance?

The distance between two points is the hypotenuse of a right triangle whose legs are the horizontal and vertical differences.

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