Mathematics · Study Guide

The Pythagorean Theorem: Statement, Proof, and Worked Examples

The Pythagorean theorem describes a fixed relationship between the three sides of any right triangle. It's one of the oldest theorems in mathematics and still one of the most useful — it shows up in geometry, trigonometry, physics, and everyday distance calculations.

The Statement

In a right triangle, let $a$ and $b$ be the two legs (the sides forming the right angle) and $c$ be the hypotenuse (the longest side, opposite the right angle). The theorem states:

a^2 + b^2 = c^2

This ONLY works for right triangles — applying it to any other triangle gives a wrong answer.

Why It's True (A Real Proof)

Here's one classic way to see why the theorem holds, using areas instead of algebra alone. Take four identical copies of the right triangle (legs $a$ , $b$ ; hypotenuse $c$ ) and arrange them inside a large square of side length $(a+b)$ , with each triangle's right angle in a corner of the big square and the hypotenuses forming a smaller, tilted square in the middle.

The big square's area can be measured two different ways. Directly, its area is $(a+b)^2$ . But it's also made up of the 4 triangles plus the inner tilted square, so its area is also $4 \cdot (\frac{1}{2}ab) + c^2$ , which simplifies to $2ab + c^2$ .

Since both expressions measure the exact same area, they must be equal:

(a+b)^2 = 2ab + c^2

Expanding the left side:

a^2 + 2ab + b^2 = 2ab + c^2

Subtracting

2ab

from both sides leaves exactly the theorem:

a^2 + b^2 = c^2

Worked Example — Finding the Hypotenuse

Problem

A right triangle has legs of length

3

and

4

. Find the hypotenuse.

Set up

Using

a^2 + b^2 = c^2

with

a=3

and

b=4

3^2 + 4^2 = c^2

Simplify

9 + 16 = c^2

25 = c^2

Solve

Take the square root of both sides — remember to do this LAST, after combining the squares, not before:

c = \sqrt{25} = 5

Common Mistakes

Using the theorem on a triangle that isn't a right triangle — it only works when there's a genuine $90°$ angle.
Mixing up which side is the hypotenuse. The hypotenuse is always the side directly across from the right angle, and it's always the LONGEST side — if your answer for $c$ comes out shorter than a leg, a mistake was made.
Forgetting to take the square root at the end and leaving the answer as $c^2 = 25$ instead of $c = 5$ .
When solving for a LEG instead of the hypotenuse (e.g. given $c$ and one leg), forgetting to subtract before taking the square root — the setup becomes $a^2 = c^2 - b^2$ , not $a^2 = c^2 + b^2$ .

Practice

A right triangle has a hypotenuse of

13

and one leg of

5

. Find the other leg.

a^2 + 5^2 = 13^2

a^2 = 169 - 25 = 144

a = 12

A ladder 10 m long leans against a wall, with its base 6 m from the wall. How high up the wall does it reach?

The ladder is the hypotenuse.

6^2 + h^2 = 10^2

h^2 = 100 - 36 = 64

h = 8 \text{ m}

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