Pythagorean Theorem Calculator
Find the missing side of a right triangle using a² + b² = c².
Enter any two sides — leave the unknown side blank
What Is the Pythagorean Theorem Calculator?
The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Given any two sides of a right triangle, this calculator finds the third side and the triangle's area.
Formula
How to Use
Enter any two of the three side lengths (a, b, c) and leave the unknown field blank. The hypotenuse c is the longest side, opposite the 90° angle. Click Calculate and the missing side will be computed along with the triangle's area.
Example Calculation
Classic 3-4-5 right triangle: a = 3, b = 4, c = ? c = √(3² + 4²) = √(9 + 16) = √25 = 5 Area = (1/2) × 3 × 4 = 6 square units Find a leg: c = 13, b = 5, a = ? a = √(13² − 5²) = √(169 − 25) = √144 = 12
Understanding Pythagorean Theorem
The Pythagorean Theorem, attributed to the ancient Greek mathematician Pythagoras (c. 570–495 BC), is one of the most fundamental results in all of mathematics. Its proof has been rediscovered over 370 different times throughout history.
The theorem has countless real-world applications: carpenters use the 3-4-5 rule to ensure corners are square, GPS systems use it in 3D distance calculations, computer graphics rely on it for collision detection, and navigation systems apply it for shortest-path calculations.
In three dimensions, the distance formula extends naturally: d = √(Δx² + Δy² + Δz²). This generalization underpins the Euclidean distance metric used throughout science and engineering.
Frequently Asked Questions
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle, always opposite the 90° right angle. It is labeled c in the formula a² + b² = c².
What are Pythagorean triples?
Pythagorean triples are sets of three integers 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.
Does the theorem apply to non-right triangles?
No, the Pythagorean Theorem only applies to right triangles. For other triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
Can sides be decimal or fractional values?
Yes. The theorem works for any positive real numbers, not just integers.
What if I enter three sides?
The calculator requires exactly one blank field. If all three are filled, it will check which one to solve for, or you can verify the triangle is valid.
Related Tools
Area Calculator — Shapes & Formulas
Calculate the area of rectangles, circles, triangles, and other common shapes.…
Trigonometry Calculator — Sin, Cos, Tan
Calculate sine, cosine, tangent, and their inverses for any angle.…
Distance Calculator — 2D & 3D Points
Calculate the distance between two points in 2D or 3D space.…