Pythagorean Theorem
Calculate sides of a right triangle: a² + b² = c²
Right Triangle Calculator
Common Pythagorean Triples
Integer solutions where a² + b² = c²
About the Theorem
The Formula
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c²
Finding the Hypotenuse
c = √(a² + b²). The hypotenuse is always the longest side and is opposite the 90° angle.
Finding a Leg
a = √(c² – b²) or b = √(c² – a²). Subtract the square of the known leg from the square of the hypotenuse.
Frequently Asked Questions
How do you find the hypotenuse using the Pythagorean theorem?
Apply c = √(a² + b²). For legs a = 3 and b = 4: c = √(9 + 16) = √25 = 5. The classic 3-4-5 right triangle is a Pythagorean triple — integer solutions where a² + b² = c². Other common triples are 5-12-13 and 8-15-17.
How do you find a missing leg of a right triangle?
Rearrange the formula: a = √(c² − b²) or b = √(c² − a²). For hypotenuse c = 13 and one leg b = 5: a = √(169 − 25) = √144 = 12. The calculator has a mode selector — choose Find a or Find b to set which side is calculated automatically.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a² + b² = c². The smallest is 3-4-5. Others include 5-12-13, 8-15-17, 7-24-25, 20-21-29, and 9-40-41. Any multiple of a triple is also a triple: 6-8-10, 9-12-15, etc.
Does the Pythagorean theorem work for all triangles?
No — it applies only to right triangles (triangles with exactly one 90° angle). For non-right triangles, the law of cosines generalizes the theorem: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. When C = 90°, cos(C) = 0 and the law of cosines reduces to a² + b² = c².
What happens in the calculator if the hypotenuse entered is shorter than a leg?
The hypotenuse must always be the longest side. If you enter a hypotenuse c that is less than or equal to leg b when solving for leg a, the calculation produces a negative under the square root — which is impossible in real numbers. The calculator detects this and displays ‘invalid’ with an explanatory message.