Finding a Missing Leg

Welcome back to Solving for Unknown Sides! You have reached lesson four of seven, which means you are past the halfway mark in this course. In our last lesson, we rearranged the Pythagorean equation to isolate a missing leg, arriving at $a^2 = c^2 - b^2$. We explored why subtraction replaces addition when the unknown is a leg rather than the hypotenuse. Today, we take that rearranged equation and build a complete, repeatable procedure for calculating the missing leg's actual length — from start to finish.

Knowing that $a^2 = c^2 - b^2$ is a powerful first move, but it only tells us the value of the leg squared. That number represents an area — specifically, the area of the square built on the missing side. To recover the side length itself, we need one more operation: the square root.

Think of it this way. If someone tells you a square tile has an area of $64$ square inches, you find its side length by asking, "What number times itself gives $64$?" The answer is $8$, because $8 \times 8 = 64$. The same logic applies here. Once subtraction gives us $a^2$, taking the square root "unwraps" that area back into a length. This is the exact same finishing move we used when finding the hypotenuse in Lesson 2 — the only difference is how we arrived at the squared value (subtraction instead of addition).

Finding a missing leg follows three clear steps. The rhythm mirrors the hypotenuse procedure, but with subtraction in the middle instead of addition:

1. Square the hypotenuse and the known leg.
2. Subtract the known leg-square from the hypotenuse-square.
3. Square root the result to get the missing leg.

Written as a single formula:

That one line captures all three steps. Let's put it to work.

Suppose a right triangle has hypotenuse $c = 15$ and one leg $b = 9$. We need to find leg $a$.

Step 1 — square the known sides:

Step 2 — subtract:

Step 3 — square root:

The missing leg is 12. A quick check confirms the result: $9^2 + 12^2 = 81 + 144 = 225 = 15^2$. ✓