Finding the Hypotenuse

Welcome back to Solving for Unknown Sides! In the first lesson of this course, you focused on setting up the Pythagorean equation from a diagram — identifying which side is the hypotenuse and writing $a^2 + b^2 = c^2$ correctly. Now, in this second lesson out of seven, we are going to put that equation to work. Our goal today is simple and satisfying: given both legs of a right triangle, calculate the hypotenuse and get an actual number.

Recall that the Pythagorean Theorem relates the three sides of any right triangle:

Here, $a$ and $b$ are the legs (the two sides that form the right angle) and $c$ is the hypotenuse (the longest side, opposite the right angle). In the previous lesson we practiced writing this equation from a diagram. Now our focus shifts to solving for $c$ so we can find its actual length.

Starting from $a^2 + b^2 = c^2$, we isolate $c$ by taking the square root of both sides:

This formula is the engine behind everything we do in this lesson. Let's break it down into a clear, repeatable procedure.

Finding the hypotenuse comes down to three steps you can follow every single time:

1. Square each leg.
2. Add the two squared values together.
3. Take the square root of that sum.

Square, add, root — that's it. The order matters, and sticking to it keeps your work organized and your answers correct. Let's see the procedure in action with real numbers.

Imagine a right triangle whose legs measure $a = 3$ and $b = 4$. We want the hypotenuse $c$.

Step 1 — Square each leg.

Step 2 — Add the squares.

Step 3 — Take the square root.

The hypotenuse is 5. Notice how each step feeds cleanly into the next — we never skip ahead, and the procedure keeps things error-free.