Right Triangle Missing Side- How to Find the Length
What Is a Right Triangle?
A right triangle has one angle that measures exactly 90 degrees. The side opposite that right angle is the hypotenuse — it's always the longest side. The other two sides are called the legs.
This shape shows up everywhere. Construction, navigation, video games, architecture. If you need to find a missing side, you're working with the Pythagorean theorem.
The Pythagorean Theorem: a² + b² = c²
This is the formula. It works every time for right triangles.
a and b are the legs. c is the hypotenuse.
The rule: the square of the hypotenuse equals the sum of the squares of the other two sides.
When to Use Which Version
- Both legs missing? You need the hypotenuse. Add the squares, then take the square root.
- Hypotenuse given, one leg missing? Subtract the known leg's square from the hypotenuse's square, then take the square root.
How to Find the Missing Side: Step by Step
Example 1: Finding the Hypotenuse
You know: a = 3, b = 4
Find c.
Step 1: Square both legs
3² = 9
4² = 16
Step 2: Add them together
9 + 16 = 25
Step 3: Take the square root
√25 = 5
Answer: c = 5
Example 2: Finding a Missing Leg
You know: c = 10, a = 6
Find b.
Step 1: Square everything
10² = 100
6² = 36
Step 2: Subtract the known leg's square from the hypotenuse's square
100 - 36 = 64
Step 3: Take the square root
√64 = 8
Answer: b = 8
Quick Reference Table
| What You Know | Formula to Use | Operation |
|---|---|---|
| Both legs (a, b) | c² = a² + b² | Add, then square root |
| Hypotenuse + one leg | a² = c² - b² | Subtract, then square root |
Common Mistakes
- Using the wrong side as the hypotenuse. The hypotenuse is always opposite the 90° angle. It's also always the longest side.
- Forgetting to square root. After you add or subtract, you still need to find the square root to get the actual length.
- Adding when you should subtract. If you're finding a leg, you subtract. Only add when finding the hypotenuse.
- Rounding too early. Keep numbers exact through calculations. Round only at the end if needed.
Practice Problems
Problem 1: Legs are 5 and 12. Find the hypotenuse.
Answer: 13 (5² + 12² = 25 + 144 = 169, √169 = 13)
Problem 2: Hypotenuse is 15, one leg is 9. Find the other leg.
Answer: 12 (15² - 9² = 225 - 81 = 144, √144 = 12)
Problem 3: Legs are 8 and 15. Find the hypotenuse.
Answer: 17 (8² + 15² = 64 + 225 = 289, √289 = 17)
When You Need Decimals
Not every answer is a clean integer. Sometimes you get a decimal.
Example: legs are 6 and 7.
6² + 7² = 36 + 49 = 85
√85 ≈ 9.22
Use a calculator if the number isn't perfect. That's fine. The method stays the same.
Real World Applications
Construction workers use this to check if corners are square. Surveyors use it to measure distances. Carpenters use the 3-4-5 rule to lay out right angles on a job site.
If you're working on a project and need to know how long a diagonal support beam should be, this is the math. Same if you're calculating the distance between two points on a grid.
The Bottom Line
Find the missing side of a right triangle? Identify which side is missing. Plug into the right version of a² + b² = c². Square, add or subtract, then square root.
That's it. No fluff needed.