Unit 7 · Topic 1

The Pythagorean Theorem

Overview

For a right triangle, a² + b² = c².

Topic 1 of 2~56 min
Next: Volume of Cylinders, Cones & SpheresAll units
Unit overview

The lesson

This lesson teaches The Pythagorean Theorem. Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.

The relationship

For any right triangle with legs a and b and hypotenuse c (the side opposite the right angle): a² + b² = c².

When you study the relationship, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.

Worked example

A right triangle has legs 3 and 4. Find the hypotenuse.

  1. 1a² + b² = c²: 3² + 4² = c².
  2. 29 + 16 = 25, so c² = 25.
  3. 3c = √25 = 5.

Why this matters

The Pythagorean Theorem shows up constantly in for a right triangle, a² + b² = c². It also connects to what you will see on homework, quizzes, and the next unit in this grade.

Teachers often move fast in class. This page is here so you can pause, re-read, and practice until the idea feels familiar, not just until you have memorized a rule for one day.

Common mistakes to avoid

Rushing to the answer without writing steps. Middle-school math rewards clear work, and you catch errors earlier when steps are visible.

Mixing up similar ideas from the same topic. If two terms feel alike, make a two-column note: what is the same, what is different, and one example of each.

Key ideas from this lesson

  1. The relationship

Video walkthrough

Math Antics

The Pythagorean Theorem

Use a² + b² = c² to find a missing side.

Watch on YouTube
Khan Academy

Pythagorean Theorem Intro

Another walkthrough of the theorem with examples.

Watch on YouTube

Practice

For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.

Exercise 1

Try it yourself

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

Step-by-step solution

  1. 15² + 12² = c² → 25 + 144 = 169.
  2. 2c² = 169, so c = √169 = 13.
  3. 3Hypotenuse: 13.

Stuck on The Pythagorean Theorem? Re-read the lesson section above, then try this exercise again without peeking.

Exercise 2

Try it yourself

A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

Step-by-step solution

  1. 113² − 5² = b² → 169 − 25 = 144.
  2. 2b = √144 = 12.
  3. 3Missing leg: 12.

Stuck on The Pythagorean Theorem? Re-read the lesson section above, then try this exercise again without peeking.

Exercise 3

Try it yourself

Do sides 7, 24, and 25 form a right triangle?

Step-by-step solution

  1. 17² + 24² = 49 + 576 = 625.
  2. 225² = 625.
  3. 3Yes, it is a right triangle.

Stuck on The Pythagorean Theorem? Re-read the lesson section above, then try this exercise again without peeking.

Exercise 4

Try it yourself

A ladder leans against a wall. The base is 6 ft from the wall and the ladder is 10 ft long. How high does it reach?

Step-by-step solution

  1. 1Legs: 6 (ground) and h (wall); hypotenuse 10.
  2. 26² + h² = 10² → 36 + h² = 100.
  3. 3h² = 64, h = 8 ft.

Stuck on The Pythagorean Theorem? Re-read the lesson section above, then try this exercise again without peeking.

Exercise 5

Try it yourself

Find the diagonal of a rectangle with length 9 and width 40.

Step-by-step solution

  1. 19² + 40² = 81 + 1600 = 1681.
  2. 2√1681 = 41.
  3. 3Diagonal length: 41.

Stuck on The Pythagorean Theorem? Re-read the lesson section above, then try this exercise again without peeking.

Khan Academy

Pythagorean theorem

Open resource
Math Worksheets 4 Kids

Pythagorean theorem worksheets

Open resource

Quick check

Answer all questions. Retake the quiz until you feel confident before moving on.

The Pythagorean Theorem

Question 1 of 4

Easy

A right triangle has legs 6 and 8. What is the hypotenuse?

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