Pythagorean Theorem Explained with Examples

Learn the Pythagorean theorem, understand when and how to use it, and solve problems involving right triangles with step-by-step examples.

Introduction

The Pythagorean theorem is one of the most famous and useful theorems in mathematics. It provides a relationship between the sides of a right triangle, making it essential for geometry, trigonometry, and many real-world applications.

Named after the ancient Greek mathematician Pythagoras, this theorem has been used for thousands of years and remains fundamental in modern mathematics and engineering.

The Pythagorean Theorem

a² + b² = c²

In a right triangle, the square of the hypotenuse equals the sum of squares of the legs.

What is the Pythagorean Theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of squares of the other two sides (legs).

Formula: a² + b² = c²

Where:

  • a and b are the legs (the two shorter sides)
  • c is the hypotenuse (the longest side, opposite the right angle)

When Does It Apply?

The Pythagorean theorem applies only to right triangles (triangles with one 90-degree angle).

It does not work for acute, obtuse, or equilateral triangles. Always verify that you're working with a right triangle before applying this theorem.

Step-by-Step Examples

Example 1: Find the Hypotenuse

Given a right triangle with legs a = 3 and b = 4, find the hypotenuse c:

  1. Apply the formula: a² + b² = c²
  2. Substitute values: 3² + 4² = c²
  3. Calculate: 9 + 16 = c²
  4. Simplify: 25 = c²
  5. Take square root: c = √25 = 5

Answer: The hypotenuse is 5 units

This is the classic 3-4-5 right triangle!

Example 2: Find a Missing Leg

Given a right triangle with hypotenuse c = 10 and leg a = 6, find leg b:

  1. Apply the formula: a² + b² = c²
  2. Substitute values: 6² + b² = 10²
  3. Calculate: 36 + b² = 100
  4. Solve for b²: b² = 100 - 36 = 64
  5. Take square root: b = √64 = 8

Answer: The missing leg is 8 units

This forms a 6-8-10 right triangle!

🌍 Real-World Applications

Construction

Ensuring square corners and right angles

Navigation

Calculating distances between points

Engineering

Designing structures and calculating forces

Computer Graphics

Calculating distances and positions

⚠️ Common Mistakes

  • Using it for non-right triangles: The theorem only works for right triangles
  • Confusing legs and hypotenuse: The hypotenuse is always the longest side
  • Forgetting to take square root: After finding c², remember to find c
  • Using wrong formula: For missing leg, use b² = c² - a², not addition

When to Use a Calculator

Pythagorean theorem calculators are helpful when:

  • Working with decimal measurements
  • Need quick, accurate results
  • Verifying manual calculations
  • Solving complex problems with multiple steps