Pythagorean Theorem Calculator
Calculate sides of a right triangle
Step-by-Step Solution
- 1a² = 3² = 9.0000
- 2b² = 4² = 16.0000
- 3a² + b² = 25.0000
- 4c = √25.0000 = 5.0000
Pythagorean Theorem
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides. This fundamental relationship was known to ancient civilizations and is named after the Greek mathematician Pythagoras.
Common Pythagorean Triples
Any multiple of a Pythagorean triple is also a triple (e.g., 6-8-10, 9-12-15).
How to Use the Pythagorean Theorem Calculator
The Pythagorean theorem calculator finds the missing side of a right triangle when you know two sides. Enter any two sides and instantly see the third, along with a visual diagram showing your triangle to scale.
The Pythagorean Theorem
For any right triangle with legs a and b and hypotenuse c:
a² + b² = c²
The hypotenuse (c) is always the longest side, opposite the 90° angle.
Solving for Each Side
- Finding the hypotenuse: c = √(a² + b²)
- Finding a leg: a = √(c² - b²) or b = √(c² - a²)
Pythagorean Triples
Pythagorean triples are sets of three whole numbers that satisfy the theorem. Common triples include:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
Any multiple of a Pythagorean triple is also a triple (e.g., 6, 8, 10).
Distance Between Two Points
The Pythagorean theorem also calculates the distance between two points (x₁, y₁) and (x₂, y₂):
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Real-World Applications
- Construction: Checking if corners are square (3-4-5 rule)
- Navigation: Finding direct distances
- Screen sizes: Calculating diagonal measurements
- Ladders: Determining safe placement angles
Related Calculators
For more triangle calculations, use our square footage calculator. For slope calculations, try our slope calculator.
Frequently Asked Questions
For a right triangle with legs a and b and hypotenuse c: a² + b² = c². To find the hypotenuse, add the squares of the legs and take the square root. To find a leg, subtract the square of the known leg from the hypotenuse squared, then take the square root.
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². Common examples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Any multiple of a triple is also a triple (6-8-10, 9-12-15). These are useful for quick mental calculations.
No, the Pythagorean theorem only works for right triangles. For other triangles, use the Law of Cosines: c² = a² + b² - 2ab·cos(C). When angle C is 90°, cos(C) = 0, and this simplifies to the Pythagorean theorem.
The distance between two points (x₁, y₁) and (x₂, y₂) forms a right triangle. The horizontal distance is |x₂ - x₁| and vertical distance is |y₂ - y₁|. These are the legs, so distance d = √[(x₂ - x₁)² + (y₂ - y₁)²].
The 3-4-5 rule verifies right angles on job sites. Measure 3 feet along one edge, 4 feet along the other, and if the diagonal is exactly 5 feet, the corner is square (90°). Any multiple works: 6-8-10, 9-12-15, 12-16-20.
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