The Pythagorean Theorem — Formula, Proof, and Examples

The Pythagorean Theorem describes the relationship between the three sides of a right triangle, and it is also known as the Three-Square Theorem. If the two sides forming the right angle are $a$ and $b$, and the hypotenuse (the side opposite the right angle) is $c$, then:

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

Here $a$ and $b$ are the legs, $c$ is the hypotenuse, and each term is squared, not just added. The most important point to remember is that this formula can only be used with right triangles.

Why the formula holds: an area argument

This theorem says that "the sum of the squares of the two shorter sides is equal to the square of the hypotenuse." We aren't adding the lengths themselves; we are comparing values raised to the $2$ power.

If you draw a square on each side of the triangle, the combined area of the two smaller squares exactly equals the area of the largest square. The expression $a^2 + b^2 = c^2$ is simply the mathematical statement of that relationship between areas.

Using area also gives one of the most intuitive proofs. Imagine a large square with side length $a+b$, and place four identical right triangles inside it. The area of the large square is:

$(a+b)^2$

This same area is also the sum of the "area of the four triangles" and the "area of the center square." Since the area of one triangle is $\frac{ab}{2}$:

$(a+b)^2 = 4 \cdot \frac{ab}{2} + c^2$

Simplifying the right side:

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

which leaves:

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

Worked example: find the hypotenuse step by step

Let's find the hypotenuse when the two sides forming the right angle are $6$ cm and $8$ cm. Substitute into the formula:

$c = \sqrt{6^2 + 8^2}$

Calculate inside the root first:

$c = \sqrt{36 + 64} = \sqrt{100} = 10$

Therefore, the hypotenuse is $10$ cm.

There are two usage scenarios worth separating. To find the hypotenuse from the two legs:

$c = \sqrt{a^2 + b^2}$

To find one of the other sides from the hypotenuse and one leg, rearrange:

$a = \sqrt{c^2 - b^2}$

Keep in mind that the hypotenuse $c$ must always be the longest side.

Direct practice and the answer check

First, try a problem where the "hypotenuse is $13$ and one side is $5$, and you need to find the other side." Use the rearranged form $a = \sqrt{c^2 - b^2}$, subtract the smaller square from the larger, and simplify.

You can also confirm a triangle is right using the converse of the theorem. With side lengths $3$, $4$, and $5$:

$3^2 + 4^2 = 9 + 16 = 25 = 5^2$

Since this holds, it is a right triangle. Note this is not about adding $3$ and $4$ to get $7$.

Calculation traps to avoid

Using it on non-right triangles

The Pythagorean Theorem only works for right triangles. Always check for a right angle before applying the formula.

Confusing which side is the hypotenuse

The hypotenuse is always the longest side, opposite the right angle. If you mix this up, the form $c^2 - b^2$ will be incorrect.

Confusing length with square

It is $a^2 + b^2 = c^2$, not $a+b=c$. A common mistake in the worked example above is computing $6+8=14$. You add the squared values, not the lengths. Understanding it as a "relationship between the areas of squares" avoids this confusion.

Forgetting to simplify square roots

Leaving an answer as $\sqrt{72}$ isn't technically wrong, but if required you should simplify:

$\sqrt{72} = 6\sqrt{2}$

Always check whether the problem asks for an integer or a simplified radical form.

Where the theorem appears

Beyond school geometry, this theorem shows up when calculating distances on a coordinate plane, finding the diagonal of a rectangle, determining the length of a ramp or ladder, and in basic architectural and surveying calculations. The distance formula for two points is essentially this theorem applied to a right triangle formed by the horizontal and vertical differences. After the practice problem above, see how this same logic connects to finding the distance between two points on a coordinate plane.

There are also integer sets that satisfy the theorem perfectly, such as $3,4,5$ and $5,12,13$. These are called Pythagorean triples and are handy for estimating answers, but the core essence is knowing how to use the formula itself.