CMClearMathAcademy

The Pythagorean Theorem

A free Pre-Algebra lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.

In a right triangle, the side opposite the right angle is the hypotenuse (c). The other two sides are legs (a and b). The Pythagorean Theorem says a^2 + b^2 = c^2. To find a missing side, substitute the known sides and solve.

What you'll learn

Why it matters: Construction (squaring a corner with a 3-4-5 layout), navigation (straight-line distance from coordinates), and screen sizes (TV diagonal from width and height) all use this theorem.

Worked example

Problem. A right triangle has legs 5 and 12. Find the hypotenuse.

  1. Use a^2 + b^2 = c^2 with a=5, b=12.
  2. 25 + 144 = c^2, so c^2 = 169.
  3. c = sqrt(169) = 13.

Answer: 13

Practice problems

1. Right triangle with legs 3 and 4. Find the hypotenuse.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Right triangle with legs 3 and 4. Find the hypotenuse.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 9 + 16 = 25.
  4. sqrt(25) = 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

2. Right triangle with legs 6 and 8. Find the hypotenuse.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Right triangle with legs 6 and 8. Find the hypotenuse.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 36 + 64 = 100.
  4. sqrt(100) = 10.
  5. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

3. Right triangle with legs 9 and 12. Find the hypotenuse.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Right triangle with legs 9 and 12. Find the hypotenuse.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 81 + 144 = 225.
  4. sqrt(225) = 15.
  5. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

4. Hypotenuse 13, one leg 5. Find the other leg.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Hypotenuse 13, one leg 5. Find the other leg.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Other leg squared = 169 - 25 = 144.
  4. sqrt(144) = 12.
  5. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

5. Hypotenuse 17, one leg 8. Find the other leg.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Hypotenuse 17, one leg 8. Find the other leg.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Other leg squared = 289 - 64 = 225.
  4. sqrt(225) = 15.
  5. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

Practice this interactively with instant feedback and an AI tutor.

Practice The Pythagorean Theorem Take the free placement check

More Pre-Algebra lessons