Pythagorean Theorem Explained

The Pythagorean theorem is one of the most useful geometry formulas because it connects the three sides of a right triangle.

Quick idea

In a right triangle, the squares of the two legs add up to the square of the hypotenuse.

Steps

1. Identify the right angle.
2. Find the hypotenuse, which is opposite the right angle.
3. Substitute the known side lengths into the formula.
4. Solve for the missing side and simplify.

Worked example

Find the hypotenuse when the legs are 3 and 4.

Full solution

Step-by-step walkthrough

Step 1

Square the legs: 3 squared is 9 and 4 squared is 16.

Step 2

Add them to get 25.

Step 3

The square root of 25 is 5.

Common mistake

Only use this theorem with right triangles. It does not work for every triangle.

Practice problems

1. Find c when a = 6 and b = 8.
2. Find a when c = 13 and b = 5.
3. Find c when a = 5 and b = 12.

Answers

1. 10
2. 12
3. 13

Ask Pennpaper to explain it live

If the steps make sense but you still feel stuck, start a Pennpaper lesson and ask the tutor to draw the problem on the whiteboard. Seeing the symbols move step by step is often what makes the concept click.

Related guides

• How to Find the Area of a Triangle
• Similar Triangles Explained
• Congruent Triangles: SSS, SAS, ASA, AAS