Pythagorean Theorem Calculator

Calculate the hypotenuse, find missing sides, and check for 3-4-5 triangles with our free Pythagorean theorem calculator

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What is the Pythagorean Theorem?

The Pythagorean theorem is a fundamental principle in geometry that relates the three sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is named after the ancient Greek mathematician Pythagoras and is one of the most well-known and widely used theorems in mathematics.

The Pythagorean Formula

The theorem is expressed as:

a^2 + b^2 = c^2

Applications

The Pythagorean theorem is used in many real-world applications, including construction (ensuring corners are square), navigation (calculating distances), computer graphics (calculating distances between points), and engineering (determining lengths and angles in structures).

Examples of Pythagorean Theorem Calculations

Here are several everyday situations that show how easy it is to use our Pythagorean theorem calculator.

Example 1: Classic 3-4-5 Triangle

A right triangle has sides of 3 and 4 units. What is the length of the hypotenuse?

You can use our calculator by entering:

What is the hypotenuse?

The calculator returns: 5 units

Given: a = 3, b = 4
Using Pythagorean theorem: c² = a² + b²
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5

The formula used is:

c = √(3^2 + 4^2) = √(9 + 16) = √(25) = 5

Example 2: Finding Missing Side

A right triangle has a hypotenuse of 10 units and one side of 6 units. What is the length of the other side?

You can use our calculator by entering:

What is the length of the other side?

The calculator returns: 8 units

Given: a = 6, c = 10
Using Pythagorean theorem: b² = c² - a²
b² = 10² - 6² = 100 - 36 = 64
b = √64 = 8

The formula used is:

b = √(c^2 - a^2) = √(10^2 - 6^2) = √(100 - 36) = √(64) = 8

Example 3: UK Ladder Against Wall

A 13-foot ladder is placed against a wall in a London building. The base of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?

You can use our calculator by entering:

How high does the ladder reach?

The calculator returns: 12 feet

Given: hypotenuse (ladder) = 13 ft, base = 5 ft
Using Pythagorean theorem: height² = ladder² - base²
height² = 13² - 5² = 169 - 25 = 144
height = √144 = 12 feet

The formula used is:

h = √(13^2 - 5^2) = √(169 - 25) = √(144) = 12

Example 4: UK Building Diagonal

A rectangular room in a Manchester building is 9 metres wide and 12 metres long. What is the length of the diagonal?

You can use our calculator by entering:

What is the diagonal length?

The calculator returns: 15 metres

Given: width = 9 m, length = 12 m
Using Pythagorean theorem: diagonal² = width² + length²
diagonal² = 9² + 12² = 81 + 144 = 225
diagonal = √225 = 15 metres

The formula used is:

d = √(9^2 + 12^2) = √(81 + 144) = √(225) = 15

Example 5: Distance Between Points

Two points on a coordinate plane are at (0, 0) and (6, 8). What is the distance between them?

You can use our calculator by entering:

What is the distance?

The calculator returns: 10 units

Given: point 1 = (0, 0), point 2 = (6, 8)
Using distance formula (based on Pythagorean theorem):
distance² = (6-0)² + (8-0)² = 36 + 64 = 100
distance = √100 = 10 units

The formula used is:

d = √((6-0)^2 + (8-0)^2) = √(36 + 64) = √(100) = 10

Example 6: UK TV Screen Diagonal

A TV screen from Currys is 16 inches wide and 12 inches tall. What is the diagonal measurement?

You can use our calculator by entering:

What is the diagonal?

The calculator returns: 20 inches

Given: width = 16 in, height = 12 in
Using Pythagorean theorem: diagonal² = width² + height²
diagonal² = 16² + 12² = 256 + 144 = 400
diagonal = √400 = 20 inches

The formula used is:

d = √(16^2 + 12^2) = √(256 + 144) = √(400) = 20

Example 7: Ramp Length

A ramp needs to rise 3 feet and extend 4 feet horizontally. How long is the ramp?

You can use our calculator by entering:

What is the ramp length?

The calculator returns: 5 feet

Given: vertical rise = 3 ft, horizontal run = 4 ft
Using Pythagorean theorem: ramp² = rise² + run²
ramp² = 3² + 4² = 9 + 16 = 25
ramp = √25 = 5 feet

The formula used is:

r = √(3^2 + 4^2) = √(9 + 16) = √(25) = 5

Example 8: Scaled 3-4-5 Triangle

A triangle has sides of 6, 8, and 10 units. Is it a 3-4-5 triangle?

You can use our calculator by entering:

Is this a 3-4-5 triangle?

The calculator returns: Yes, scaled by 2

Given: sides = 6, 8, 10
Check if it's a right triangle: 6² + 8² = 36 + 64 = 100, 10² = 100
Since 6² + 8² = 10², it's a right triangle
Check scale: 6/3 = 2, 8/4 = 2, 10/5 = 2
Yes, this is a 3-4-5 triangle scaled by 2

The formula used is:

6:8:10 = 3 × 2:4 × 2:5 × 2

Example 9: Roof Pitch

A roof rises 4 feet for every 12 feet of horizontal run. What is the length of the roof?

You can use our calculator by entering:

What is the roof length?

The calculator returns: 12.65 feet

Given: rise = 4 ft, run = 12 ft
Using Pythagorean theorem: roof² = rise² + run²
roof² = 4² + 12² = 16 + 144 = 160
roof = √160 ≈ 12.65 feet

The formula used is:

r = √(4^2 + 12^2) = √(16 + 144) = √(160) ≈ 12.65

Example 10: Cable Length

A cable needs to span 15 feet horizontally and 20 feet vertically. How long is the cable?

You can use our calculator by entering:

What is the cable length?

The calculator returns: 25 feet

Given: horizontal = 15 ft, vertical = 20 ft
Using Pythagorean theorem: cable² = horizontal² + vertical²
cable² = 15² + 20² = 225 + 400 = 625
cable = √625 = 25 feet

The formula used is:

c = √(15^2 + 20^2) = √(225 + 400) = √(625) = 25

Summary

These examples show how our Pythagorean theorem calculator can handle:

  • •Finding the hypotenuse when two sides are known
  • •Finding a missing side when the hypotenuse and one side are known
  • •Checking if a triangle is a right triangle
  • •Verifying 3-4-5 triangles and their scaled versions

You can adapt any real-life situation involving right triangles to one of these patterns and let the calculator do the work for you.

How to Use the Pythagorean Theorem Calculator

  1. Select the type of pythagorean calculation you need.
  2. Enter the known side lengths.
  3. View the calculated result and step-by-step breakdown.

Frequently asked questions

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².

When can I use the Pythagorean theorem?

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

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About this tool

Published by Online Tools Center

Last updated:

Check the units, assumptions and rounding before using a result. Health and financial estimates are educational and do not replace individual professional advice.

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