MulticalcPythagorean Theorem Calculator
Solve for any side of a right triangleSelect what to find and enter the known values.
Pythagorean Theorem Guide
What is the Pythagorean Theorem?
The **Pythagorean Theorem** is a fundamental principle in geometry that describes the relationship between the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The Formula
a² + b² = c²
where c is the hypotenuse, and a and b are the other two sides
How to Use the Calculator
1. Finding the Hypotenuse (c)
When you know both legs (a and b), use: c = √(a² + b²)
Example: If a = 3 and b = 4, then c = √(9 + 16) = √25 = 5
2. Finding a Leg (a or b)
When you know the hypotenuse and one leg, use: a = √(c² - b²) or b = √(c² - a²)
Example: If c = 5 and b = 4, then a = √(25 - 16) = √9 = 3
Real-World Applications
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Construction: Calculate diagonal bracing and roof slopes
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Navigation: Find shortest distances and bearings
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Engineering: Design structures and calculate forces
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Computer Graphics: Calculate distances in 2D and 3D space
Quick Tips
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Right Triangle
Only
The Pythagorean theorem ONLY works for right triangles (one 90° angle). -
Hypotenuse
The hypotenuse is always the longest side and is opposite the right angle. -
Pythagorean Triples
Common integer solutions: (3,4,5), (5,12,13), (8,15,17), (7,24,25).
Common Pythagorean Triples
- 3-4-5 triangle
- 5-12-13 triangle
- 8-15-17 triangle
- 7-24-25 triangle
- Multiples also work: 6-8-10, 9-12-15, etc.
Right Triangle Geometry
The Pythagorean theorem is one of the most important formulas in mathematics. Named after the Greek mathematician Pythagoras, it's been used for over 2,500 years!
Key Concepts
Learn More
Pythagorean Theorem Resources:
Did You Know?
The Pythagorean theorem has over 400 different proofs! Even U.S. President James Garfield published his own proof in 1876 before becoming president.