Free Interactive Geometry Tool

Pythagorean Theorem

Use interactive right-triangle diagrams to identify the hypotenuse, substitute known side lengths, solve for a missing leg or hypotenuse, and verify each answer.

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Learn the Pythagorean Theorem Step by Step

Review right triangles, legs, the hypotenuse, square roots, coordinate-plane applications, and complete worked examples.

Read the Pythagorean Theorem Guide

How It Works

Connect the Diagram to the Equation

The Pythagorean theorem relates the two legs of a right triangle to its hypotenuse. Identify each side, substitute the known lengths, and solve one step at a time.

1

Identify the Hypotenuse

Find the side opposite the right angle. That side is always labeled c.

2

Substitute the Lengths

Place the leg lengths in a and b and the hypotenuse length in c.

3

Solve the Equation

Square the known values, isolate the missing square, and take the square root.

4

Check the Diagram

Confirm that the hypotenuse is the longest side and that the completed equation is true.

Interactive Practice

Solve Each Right Triangle

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Problems 0
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Find the Hypotenuse

Medium

Use the two leg lengths to find the hypotenuse.

A right triangle has legs measuring 6 units and 8 units. Find the hypotenuse.
The side opposite the right angle is the hypotenuse.
0 steps completed Step 1 of 6

Interactive Diagram

Right Triangle

Study the sides

The square marker identifies the right angle.

Work One Step at a Time

Identify the Hypotenuse

Current Step

Current Problem

Find c when a = 6 and b = 8.
Pythagorean Theorem a² + b² = c²
Start a = 6, b = 8, c = ?

The hypotenuse is opposite the right angle.

Keep This Nearby

Pythagorean Theorem Reference

The hypotenuse is always c.

a² + b² = c²
The Legs The two sides that form the right angle are labeled a and b. Legs: a and b
The Hypotenuse The side opposite the right angle is the hypotenuse and is labeled c. Hypotenuse: c
Find the Hypotenuse Add the squares of the two legs, then take the square root. c = √(a² + b²)
Find a Missing Leg Subtract the known leg’s square from the hypotenuse’s square. a = √(c² − b²)
Check the Longest Side The hypotenuse must be longer than either individual leg. c > a and c > b
Exact and Approximate Answers Keep a square root exact unless the problem asks for a decimal approximation. √20 = 2√5 ≈ 4.47

Key Reminders

Avoid Common Pythagorean Theorem Mistakes

Identify the hypotenuse before substituting values and keep the equation balanced while solving.

Use Only Right Triangles

The Pythagorean theorem applies only when the triangle contains a 90-degree angle.

a² + b² = c²

Identify c Correctly

The hypotenuse is opposite the right angle, not simply the side that looks slanted.

c = hypotenuse

Square Before Adding

Calculate each square separately before adding or subtracting the results.

6² + 8² = 36 + 64

Take the Final Square Root

If c² = 100, the positive side length is c = 10, not 100.

c = √100 = 10

Keep Learning

Understand Why the Formula Works

Review right-triangle vocabulary, missing-side equations, square roots, coordinate applications, and complete worked examples in the companion guide.

Read the Pythagorean Theorem Guide

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