Does the Pythagorean theorem work for all triangles?

No, it only works for right-angled triangles. If the triangle does not have a 90-degree angle, you cannot use $a^2 + b^2 = c^2$ to find missing sides. For other triangles, you would need advanced tools like the Law of Sines or the Law of Cosines.

Does it matter which leg I call 'a' and which I call 'b'?

Not at all. Because you are just adding the squares of the legs together, the order doesn't matter (since $3^2 + 4^2$ is the exact same as $4^2 + 3^2$). Just ensure the hypotenuse is always $c$.

What if the square root isn't a whole number?

That is completely normal! Many missing sides will be decimals. Depending on what your teacher asks for, you can either leave the answer as a radical (like $\\sqrt{17}$) or use a calculator to find the decimal approximation (like 4.12).

How do you find a missing side with the Pythagorean theorem?

The Pythagorean theorem tells us that in a right-angled triangle, the square of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides. You can use it to find any missing side as long as you know the lengths of the other two.

Think of it like a strict recipe for a specific type of triangle. If you have a right triangle, the three sides are always locked into this mathematical relationship. If one piece of the puzzle is missing, the formula lets you work backward to figure out exactly what it must be.

The formula and its parts

The Pythagorean theorem is written as the equation $a^2 + b^2 = c^2$ . In this formula, $a$ and $b$ represent the lengths of the legs of the right triangle. The legs are the two shorter sides that meet to form the 90-degree right angle.

The letter $c$ always represents the hypotenuse. The hypotenuse is the longest side of the right triangle, and it is always directly across from the right angle. Identifying $c$ correctly is the most important step before doing any math.

Finding a missing hypotenuse

If you know the lengths of both legs ( $a$ and $b$ ) and need to find the hypotenuse ( $c$ ), you plug the leg lengths into the left side of the equation. Square each number, and then add those two squares together.

This gives you the value of $c^2$ . To find the actual length of the hypotenuse, $c$ , you must take the square root of that sum. For example, if $a^2 + b^2 = 25$ , then $c = 5$ because the square root of 25 is 5.

Finding a missing leg

Sometimes you already know the hypotenuse ( $c$ ) and one leg (let's say $a$ ), but you need to find the other leg ( $b$ ). You still use $a^2 + b^2 = c^2$ , but you have to rearrange it to solve for the missing piece.

To do this, square the hypotenuse and square the known leg. Then, subtract the leg's square from the hypotenuse's square. This gives you $b^2$ . Finally, take the square root of that difference to find the length of leg $b$ .

Where students usually slip up

The most common mistake is forgetting the final step: taking the square root. Many students will do the addition or subtraction, find a number like 144, and write that down as the missing side length. Always look at your triangle to see if the answer makes sense. A side of 144 on a triangle with other sides of 9 and 12 is impossible!

The second most common mistake is plugging a leg value into the $c$ spot. Remember, $c$ must be the hypotenuse, which is the side opposite the right angle. It does not matter which leg is $a$ and which is $b$ , but $c$ is strictly reserved for the longest side.

Worked through

A right triangle has a hypotenuse of 13 centimeters and one leg that is 5 centimeters. What is the length of the other leg?

First, identify your given values. The hypotenuse is $c = 13$ . The known leg is $a = 5$ . We need to find $b$ .

Start with the formula: $a^2 + b^2 = c^2$

Plug in the known values: $5^2 + b^2 = 13^2$

Square the numbers: $25 + b^2 = 169$

Subtract 25 from both sides to isolate $b^2$ : $b^2 = 169 - 25$ $b^2 = 144$

Take the square root of both sides to find $b$ : $b = \\sqrt{144}$ $b = 12$

The missing leg is 12 centimeters long.

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Where this comes from: OpenStax Prealgebra 2e: Chapter 9, Math Models and Geometry · Khan Academy: 8th grade, Pythagorean theorem unit