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

In this formula, $a$ and $b$ are the two legs and $c$ is the hypotenuse, which is the side opposite the right angle and always the longest side. Use the formula only when the triangle has a right angle.

Why the theorem is true

The theorem is easier to trust 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 relationship is $a^2 + b^2 = c^2$ rather than $a + b = c$.

One classic proof makes this precise. Start 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

The four triangles together have area

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

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

Expand and simplify:

Worked example: find the hypotenuse

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

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

Practice with a check

Now try your own version with legs $5$ and $12$. Square each leg, add, and take the square root the same way as the worked example. If you get $13$, you set the problem up correctly, and the answer passing the "longer than either leg" check confirms it.

For a useful next step, work a similar problem with the Distance Formula to see how the same idea applies on the coordinate plane.

Calculation traps with $a^2 + b^2 = c^2$

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

Another 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 $100$. Take the square root before reporting the answer.

A final trap is mixing 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.