Pythagorean Theorem Problems- Practice Exercises
What You're Getting Into
You're here to practice the Pythagorean Theorem. Not to read fluff about how math is beautiful or how geometry will change your life. This is straightforward: problems, solutions, and the mistakes most people make.
If you don't know what a² + b² = c² means, stop here. Go learn that first. This article assumes you've got the basics down.
The Formula You Already Know
In a right triangle:
- a and b are the legs (the sides touching the right angle)
- c is the hypotenuse (the longest side, opposite the right angle)
The theorem states: a² + b² = c²
That's it. Everything below is just applying this one equation to different situations.
Practice Problems with Solutions
Work through these. Check your answers only after you've tried. No peeking.
Problem 1: Find the Hypotenuse
A right triangle has legs of 3 and 4. Find the hypotenuse.
Solution:
3² + 4² = c²
9 + 16 = c²
25 = c²
c = 5
Problem 2: Find a Missing Leg
A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.
Solution:
5² + b² = 13²
25 + b² = 169
b² = 144
b = 12
Problem 3: Word Problem
A ladder leans against a wall. The base is 6 feet from the wall. The top touches the wall at 8 feet high. How long is the ladder?
Solution:
6² + 8² = c²
36 + 64 = c²
100 = c²
c = 10 feet
Problem 4: Distance Between Points
Find the distance between points (0, 0) and (7, 24).
Solution:
Δx = 7, Δy = 24
7² + 24² = c²
49 + 576 = c²
625 = c²
c = 25
Problem 5: Multi-Step
A rectangle is 9 cm long and 12 cm wide. What's the diagonal length?
Solution:
The diagonal splits the rectangle into two right triangles.
9² + 12² = c²
81 + 144 = c²
225 = c²
c = 15 cm
Common Mistakes
- Forgetting to square root. If you get c² = 64, c is not 64. c = 8. The square root is not optional.
- Using the wrong side for c. The hypotenuse is always the longest side. Always. If your answer is smaller than one of your legs, something's wrong.
- Adding instead of squaring. 3² + 4² is NOT 3 + 4. It's 9 + 16. Square first, then add.
- Mixing up which side is missing. If you're finding a leg, subtract. If you're finding the hypotenuse, add the squares of both legs.
Quick Reference: Problem Types
| Problem Type | Given | Find | Method |
|---|---|---|---|
| Find hypotenuse | a = 3, b = 4 | c | a² + b² = c² |
| Find a leg | c = 10, b = 6 | a | c² - b² = a² |
| Distance formula | Two coordinate points | Distance | Δx² + Δy² = d² |
| Rectangle diagonal | Length and width | Diagonal | l² + w² = d² |
Getting Started: Your Practice Plan
Stop reading. Start doing.
- Generate 10 problems. Use random numbers for legs (between 1-20). Calculate hypotenuse. Mix in some where you hide one leg instead.
- Check your work. Plug your answer back into a² + b². Does it equal c²? If not, you messed up.
- Time yourself. Aim for under 2 minutes per problem once you've got the hang of it.
- Mix in word problems. Ladders, stairs, diagonals. Real applications force you to identify which sides are which.
When You're Stuck
If you can't solve a problem:
- Draw the triangle. Label the sides. Most people skip this and get confused.
- Identify which side is the hypotenuse. Circle it. This single step clears up 90% of errors.
- Write the formula with your numbers plugged in. Then solve step by step.
That's the process. There's no trick. Just apply the formula, do the math, and check your work.