The Pythagorean Theorem

The Pythagorean theorem says that in any right triangle, the two shorter sides and the longest side are locked together by one equation: $a^2 + b^2 = c^2$ . The two legs are $a$ and $b$ , and $c$ is the hypotenuse — the side across from the right angle. Square the legs, add them, and you get the square of the hypotenuse, every time.

That one equation lets you find any side of a right triangle when you know the other two, which is why it shows up everywhere: distances on a coordinate plane, ladders against walls, diagonals of rectangles, and half the problems in a geometry course. The only setup skill you need is identifying which side is the hypotenuse.

What the theorem says

In the triangle below, the legs $a$ and $b$ meet at the right angle, and the hypotenuse $c$ sits across from it. The hypotenuse is always the longest side — if your computed hypotenuse comes out shorter than a leg, something went wrong.

The theorem only works on right triangles. No right angle, no $a^2 + b^2 = c^2$ . It doesn't matter which leg you call $a$ and which you call $b$ , but $c$ must be the hypotenuse.

Finding the hypotenuse vs. finding a leg

If the legs are given and the hypotenuse is missing, the setup is direct: square both legs, add, and take the square root. Legs of $3$ and $4$ give $9 + 16 = 25$ , so $c = \sqrt{25} = 5$ .

If the hypotenuse is given and a leg is missing, the hypotenuse still goes in the $c$ spot — which means you subtract instead of add. With a hypotenuse of $10$ and a leg of $6$ , the setup is $6^2 + b^2 = 10^2$ , so $b^2 = 100 - 36 = 64$ and $b = 8$ .

The most common mistake is putting a given hypotenuse in a leg spot and adding when you should subtract. Before substituting anything, find the right angle and mark the side across from it as $c$ .

Pythagorean triples and messy answers

Some whole-number side combinations satisfy the theorem exactly: $3$ - $4$ - $5$ , $6$ - $8$ - $10$ , $5$ - $12$ - $13$ , and $8$ - $15$ - $17$ are the ones worth memorizing. Multiples work too — since $3$ - $4$ - $5$ is a triple, so is $9$ - $12$ - $15$ . Spotting a triple saves you the arithmetic entirely.

Most right triangles are not triples, and that's fine — the answer is just a square root. Legs of $4$ and $7$ give $c^2 = 16 + 49 = 65$ , so $c = \sqrt{65}$ . Leave it as a simplified radical unless the problem asks for a decimal, in which case round: $\sqrt{65} \approx 8.06$ .

Worked examples

Example 1: finding the hypotenuse

A right triangle has legs of $3$ and $4$ . Find the hypotenuse.

Write the theorem

a^2 + b^2 = c^2

Substitute the legs

3^2 + 4^2 = c^2

Square and add

9 + 16 = 25

Take the square root

c = \sqrt{25} = 5

Answer: $c = 5$

Example 2: finding a missing leg

A right triangle has a hypotenuse of $10$ and one leg of $6$ . Find the other leg.

Write the theorem — the hypotenuse

10

goes in the

c

spot

6^2 + b^2 = 10^2

Square the known sides

36 + b^2 = 100

Subtract

36

from both sides

b^2 = 64

Take the square root

b = 8

Answer: $b = 8$

Example 3: a 5-12-13 triple

A right triangle has legs of $5$ and $12$ . Find the hypotenuse.

Write the theorem

5^2 + 12^2 = c^2

Square and add

25 + 144 = 169

Take the square root

c = \sqrt{169} = 13

Recognize it:

5

12

13

is a Pythagorean triple worth remembering

Answer: $c = 13$

Example 4: a square-root answer

A right triangle has legs of $4$ and $7$ . Find the exact length of the hypotenuse.

Write the theorem

4^2 + 7^2 = c^2

Square and add

16 + 49 = 65

Take the square root

c = \sqrt{65}

65 = 5 \cdot 13

has no perfect-square factor, so the radical is already simplified; as a decimal,

c \approx 8.06

Answer: $c = \sqrt{65} \approx 8.06$

Try one yourself

Common questions

How do I know which side is the hypotenuse?

It's the side directly across from the right angle, and it's always the longest side. The two sides that form the right angle are the legs. Find the little square marking the right angle first, then look at the side that doesn't touch it.

Does the theorem work on every triangle?

No — only right triangles. In fact, that's a test: if the three sides of a triangle satisfy $a^2 + b^2 = c^2$ with $c$ as the longest side, the triangle must be a right triangle. That reverse direction is called the converse of the Pythagorean theorem.

What if my answer isn't a whole number?

That's normal. Take the square root and leave it as a simplified radical like $\sqrt{65}$ , or round to a decimal if the problem asks. Only special side combinations called Pythagorean triples, like $3$ - $4$ - $5$ and $5$ - $12$ - $13$ , come out whole.

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The Pythagorean Theorem

What the theorem says

Finding the hypotenuse vs. finding a leg

Pythagorean triples and messy answers

Worked examples

Example 1: finding the hypotenuse

Example 2: finding a missing leg

Example 3: a 5-12-13 triple

Example 4: a square-root answer

Try one yourself

Common questions

Keep learning

Quadratic Formula

The Slope Formula

Factoring Trinomials (a = 1)