Pythagorean Theorem Calculator
Calculate missing sides of right triangles using the Pythagorean theorem. Find hypotenuse, legs, or verify if three sides form a right triangle with step-by-step solutions.
Pythagorean Theorem Calculator
Input Values
Find the hypotenuse when you know both legs using a² + b² = c².
Results
Triangle Sides
Leg a:3.000 cm
Leg b:4.000 cm
Hypotenuse c:5.000 cm
Area
6.000 cm²
A = ½ab
Perimeter
12.000 cm
P = a + b + c
Triangle Angles
Angle A:36.870°
Angle B:53.130°
Angle C:90.000°
Powered by Free-Calculator.com
What is Pythagorean Theorem Calculator?
The Pythagorean theorem calculator helps you find missing sides of right triangles and verify if three given sides form a right triangle. This fundamental geometric principle states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
The Pythagorean Theorem
The theorem states: a² + b² = c²
- a and b: The two legs (shorter sides) of the right triangle
- c: The hypotenuse (longest side, opposite the right angle)
Calculation Modes
- Find Hypotenuse: Given both legs, calculate c = √(a² + b²)
- Find Leg a: Given leg b and hypotenuse, calculate a = √(c² - b²)
- Find Leg b: Given leg a and hypotenuse, calculate b = √(c² - a²)
- Verify Triangle: Check if three sides satisfy the Pythagorean theorem
Pythagorean Triples
Special sets of three positive integers that satisfy the theorem:
- 3-4-5: The most famous triple (3² + 4² = 9 + 16 = 25 = 5²)
- 5-12-13: Another common triple (5² + 12² = 25 + 144 = 169 = 13²)
- 8-15-17: Less common but useful (8² + 15² = 64 + 225 = 289 = 17²)
- Scaled Triples: Any multiple of a triple is also a triple (6-8-10, 9-12-15, etc.)
Practical Applications
- Construction: Ensuring square corners and right angles
- Carpentry: Measuring diagonal cuts and frame construction
- Surveying: Land measurement and boundary determination
- Navigation: Distance calculations and GPS systems
- Architecture: Roof pitch, ramp design, and structural analysis
- Engineering: Force calculations and vector analysis
Historical Background
- Ancient Origins: Known to Babylonians and Egyptians over 4000 years ago
- Pythagoras: Greek mathematician (c. 570-495 BC) who provided the first proof
- Universal Principle: One of the most important theorems in mathematics
- Modern Relevance: Foundation for trigonometry, calculus, and physics
Verification and Accuracy
- Tolerance: Calculator uses 0.0001 tolerance for verification
- Floating Point: Small computational errors may occur
- Practical Use: Real-world measurements should account for measurement errors
- Triangle Inequality: Automatically checks if sides can form a valid triangle
FAQ - Pythagorean Theorem Calculator
The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse. It only applies to right triangles (triangles with one 90° angle).