The Pythagorean Theorem: Formula, Proof, and Real-Life Applications

What is the Pythagorean theorem?

The Pythagorean theorem is the most famous result in elementary geometry. It says that in any right triangle — a triangle with one $90^\circ$ angle — the squares on the two shorter sides add up to the square on the longest side.

If the two legs (the sides next to the right angle) have lengths $a$ and $b$, and the hypotenuse (the side opposite the right angle, always the longest side) has length $c$, then

That single line lets you find any one side of a right triangle when you know the other two.

A first example

Suppose a right triangle has legs of length $3$ and $4$. What is the hypotenuse?

1. Plug into the formula: $3^2 + 4^2 = c^2$.
2. Compute: $9 + 16 = 25$, so $c^2 = 25$.
3. Take the positive square root: $c = \sqrt{25} = 5$.

So the hypotenuse is $5$. The triple $(3, 4, 5)$ is the most famous example of a Pythagorean triple — three whole numbers that satisfy $a^2 + b^2 = c^2$.

Why does it work? A visual proof

There are over a hundred known proofs of the Pythagorean theorem. Here is the cleanest one to picture in your head.

Take a square with side length $a + b$ and arrange four copies of the same right triangle (legs $a$, $b$, hypotenuse $c$) inside it in two different ways:

• Arrangement 1: place the triangles in the four corners so the empty middle is a square of side $c$. The empty area is $c^2$.
• Arrangement 2: rearrange the same four triangles so the empty space splits into a square of side $a$ plus a square of side $b$. The empty area is $a^2 + b^2$.

Both arrangements use exactly the same four triangles inside the same big square, so the empty area must be the same. Therefore $a^2 + b^2 = c^2$.

Finding any side of a right triangle

The formula has three letters, so there are three "missing-piece" problems.

Find the hypotenuse. Given legs $a$ and $b$:

Find a missing leg. Given the hypotenuse $c$ and one leg $a$:

Check whether a triangle is a right triangle. Plug the three side lengths into $a^2 + b^2$ and $c^2$ (with $c$ as the longest side). If the two are equal, the triangle has a right angle.

Worked examples

Example 1: Missing leg. A right triangle has hypotenuse $13$ and one leg $5$. Find the other leg.

So the missing leg is $12$. Notice $(5, 12, 13)$ is another Pythagorean triple.

Example 2: Decimal answer. A right triangle has legs $5$ and $7$. What is the hypotenuse?

The exact answer is $\sqrt{74}$. The decimal $8.6$ is a rounded approximation.

Example 3: Real-world ladder. A 10-foot ladder leans against a wall. Its base is 6 feet from the wall. How high up the wall does the ladder reach?

The ladder, the wall, and the ground form a right triangle, with the ladder as the hypotenuse:

Example 4: Is this a right triangle? A triangle has sides $7$, $24$, and $25$. Is it a right triangle?

Check the longest side as $c$: $7^2 + 24^2 = 49 + 576 = 625$, and $25^2 = 625$. They match, so yes, this is a right triangle.

Pythagorean triples worth memorizing

A few small whole-number triples come up constantly. Recognizing them lets you answer a problem without reaching for a calculator.

| Triple $(a, b, c)$ | Notes | |---|---| | $(3, 4, 5)$ | The classic. Multiples like $(6, 8, 10)$ and $(9, 12, 15)$ also work. | | $(5, 12, 13)$ | Common in textbook problems. | | $(8, 15, 17)$ | Less common but useful. | | $(7, 24, 25)$ | Shows up in standardized tests. |

Any positive multiple of a Pythagorean triple is also a Pythagorean triple, because multiplying every side by the same factor scales the squares by that factor squared on both sides of the equation.

Distance between two points

The Pythagorean theorem also gives the distance formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane. Drawing a horizontal and a vertical leg from one point to the other forms a right triangle with legs $|x_2 - x_1|$ and $|y_2 - y_1|$. The distance between the points is the hypotenuse:

So the distance formula is just the Pythagorean theorem in disguise.

Common mistakes

• Using the formula on a non-right triangle. It only works when one angle is exactly $90^\circ$.
• Confusing the hypotenuse with a leg. The hypotenuse is always the longest side, opposite the right angle. It plays the role of $c$ in the formula.
• Forgetting to square root. $a^2 + b^2 = c^2$ gives you $c^2$, not $c$. Take the square root at the end.
• Subtracting in the wrong order. When solving for a leg, the formula is $b = \sqrt{c^2 - a^2}$, not $\sqrt{a^2 - c^2}$.