Pythagoras Triangle Calculator
Calculate missing sides of right triangles using the Pythagorean theorem (a² + b² = c²)
Calculate Right Triangle
One of the perpendicular sides
One of the perpendicular sides
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
Formula Reference
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