Pythagorean Theorem — Formula, Proof & Examples

The Pythagorean theorem says that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the legs are $a$ and $b$ and the hypotenuse is $c$, then

$$a^2 + b^2 = c^2$$

Use this formula only when the triangle has a right angle. The hypotenuse is the side opposite that right angle, and it is always the longest side.

What the formula means

The theorem is easier to remember if you think in areas, not just side lengths. Build a square on each side of a right triangle. The area of the square on side $c$ matches the combined areas of the squares on sides $a$ and $b$.

That is why the side lengths are squared. The theorem is comparing square areas, which is why the relationship is $a^2 + b^2 = c^2$ instead of $a + b = c$.

Why the Pythagorean theorem is true

One classic proof starts with a large square of side length $a + b$. Put four identical right triangles inside it so that their hypotenuses form a smaller inner square.

The large square has area

$$(a+b)^2$$

The four triangles together have area

$$4 \left( \frac{1}{2}ab \right) = 2ab$$

The inner square has side length $c$, so its area is

$$c^2$$

Since the large square is made of those four triangles plus the inner square,

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

Expand and simplify:

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

Worked example: find the hypotenuse

Suppose a right triangle has legs $6$ and $8$. To find the hypotenuse $c$, substitute those values into the theorem:

$$6^2 + 8^2 = c^2$$ $$36 + 64 = c^2$$ $$100 = c^2$$ $$c = 10$$

So the hypotenuse is $10$. That answer makes sense because the hypotenuse should be longer than either leg.

Common mistakes with $a^2 + b^2 = c^2$

The most common mistake is using the theorem on a triangle that is not right. The formula needs a $90^\circ$ angle.

Another mistake is putting the wrong side in place of $c$. The hypotenuse is always opposite the right angle, and it is always the longest side.

Students also sometimes stop too early. If you get $c^2 = 100$, the side length is $c = 10$, not $100$.

Some students also mix up $a^2 + b^2 = c^2$ with $(a+b)^2 = c^2$. Those are not the same expression.

When to use the Pythagorean theorem

Use the theorem when two lengths meet at a right angle and you need the direct distance across from that angle. Common cases include the diagonal of a rectangle, the straight-line distance between two points, and basic construction or surveying layouts.

It is also useful for checking whether a triangle is right. If side lengths of a triangle satisfy $a^2 + b^2 = c^2$ with $c$ as the longest side, then the triangle is a right triangle.

Try a similar problem

Try your own version with legs $5$ and $12$. If you get $13$, you set the problem up correctly.

If you want a useful next step, explore a similar problem with the Distance Formula to see how the same idea works on the coordinate plane.