Pythagorean Theorem: Hypotenuse and Legs
Gianluca
December 3, 2025
The Pythagorean Theorem is a fundamental relation in geometry connecting the three sides of a right triangle.
The Formula
$$a^2 + b^2 = c^2$$- \(a\) and \(b\): The Legs (the two sides that form the right angle).
- \(c\): The Hypotenuse (the longest side, opposite the right angle).
Step-by-Step Example
Find the length of the missing leg \(a\) in a right triangle where the hypotenuse is \(c = 10\) and the other leg is \(b = 8\).
Step 1: Identify the sides
- \(b = 8\) (Leg)
- \(c = 10\) (Hypotenuse)
- \(a = ?\) (Leg)
Step 2: Plug into the formula
$$a^2 + 8^2 = 10^2$$Step 3: Solve for \(a^2\)
$$a^2 + 64 = 100$$$$a^2 = 100 - 64$$
$$a^2 = 36$$
Step 4: Take the square root
$$a = \sqrt{36}$$$$a = 6$$
Solution: The missing leg is \(6\).
The most common mistake in exams is blindly applying $a^2 + b^2 = c^2$ even when you are looking for a leg (not the hypotenuse).
The Golden Rule:
- Looking for the longest side ($c$)? Add the squares: $c = \sqrt{a^2 + b^2}$
- Looking for a short side ($a$ or $b$)? Subtract the squares: $a = \sqrt{c^2 - b^2}$
The “Sanity Check”: Before finishing, look at your result. The hypotenuse must be the longest side of the triangle. If you calculated a leg and it’s longer than the hypotenuse, you accidentally added instead of subtracting.
Example: If $c=10$ and $b=8$, many students write $10^2 + 8^2 = 164 \rightarrow \sqrt{164} \approx 12.8$. Wrong! A leg (12.8) cannot be longer than the hypotenuse (10). The correct path is $10^2 - 8^2 = 36 \rightarrow \sqrt{36} = 6$.
If you struggle with moving terms in an equation, review our guide on Literal Equations and Formulas.
Why reading isn’t enough
The Pythagorean Theorem is simple, but real-world problems often involve decimals or finding a leg instead of the hypotenuse (\(a^2 = c^2 - b^2\)).
Weekzen offers a variety of problems, including word problems and complex figures, to help you visualize and apply the theorem correctly in any situation.