Pythagoras Triangle Calculator

Calculate missing sides of right triangles using the Pythagorean theorem (a² + b² = c²)

Calculate Right Triangle

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One of the perpendicular sides

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One of the perpendicular sides

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This value will be calculated

Example Calculations

Famous 3-4-5 Triangle

Given: a = 3, b = 4

Find: c = √(3² + 4²) = √(9 + 16) = √25 = 5

Verification: 3² + 4² = 9 + 16 = 25 = 5²

This is a Pythagorean triple!

Finding a Cathetus

Given: b = 6, c = 10

Find: a = √(c² - b²) = √(10² - 6²) = √(100 - 36) = √64 = 8

Verification: 8² + 6² = 64 + 36 = 100 = 10²

Another Pythagorean triple: (6, 8, 10)

Non-Integer Result

Given: a = 5, b = 7

Find: c = √(5² + 7²) = √(25 + 49) = √74 ≈ 8.602

Note: Not all right triangles have integer sides

Common Pythagorean Triples

(3, 4, 5)Basic triple
(5, 12, 13)Popular triple
(8, 15, 17)Medium triple
(7, 24, 25)Large triple
(6, 8, 10)Multiple of (3,4,5)
(9, 12, 15)Multiple of (3,4,5)

Formula Reference

a² + b² = c²
Pythagorean Theorem

Find hypotenuse:
c = √(a² + b²)

Find cathetus:
a = √(c² - b²)
b = √(c² - a²)

Area:
A = (a × b) / 2

Perimeter:
P = a + b + c

Quick Tips

The hypotenuse is always the longest side

Only works for right triangles (90° angle)

Pythagorean triples have integer sides

Used in construction, navigation, and design

Understanding the Pythagorean Theorem

What is a Pythagoras Triangle?

A Pythagoras triangle, also known as a right triangle, is any triangle that contains a 90-degree angle. The side opposite to the right angle is called the hypotenuse, and it's always the longest side. The other two sides are called catheti or legs.

The Pythagorean Theorem

The theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides: a² + b² = c². This relationship allows us to find any missing side when we know the other two.

Calculation Methods

Finding the Hypotenuse

When you know both catheti (a and b):

c = √(a² + b²)

Finding a Cathetus

When you know the hypotenuse and one cathetus:

a = √(c² - b²)

b = √(c² - a²)

Historical Context

The Pythagorean theorem is named after the ancient Greek mathematician Pythagoras (c. 570-495 BCE), although the relationship was known to earlier civilizations including the Babylonians and Chinese. The theorem has been proven in hundreds of different ways and remains one of the most important mathematical relationships.

Real-World Applications

Construction & Architecture

  • • Ensuring square corners in buildings
  • • Calculating roof slopes and angles
  • • Determining diagonal bracing lengths
  • • Foundation layout and surveying
  • • Stair design and construction

Navigation & GPS

  • • Calculating distances between points
  • • GPS triangulation methods
  • • Flight path calculations
  • • Maritime navigation
  • • Satellite positioning systems

Engineering & Design

  • • Mechanical component design
  • • Force vector calculations
  • • Screen and display sizing
  • • Antenna positioning
  • • Bridge and structure analysis