Enter any two sides of a right triangle — side a, side b, or hypotenuse c — and the Pythagorean Theorem Calculator finds the missing third side. Leave exactly one field blank, and you'll also get the triangle area calculated automatically using a² + b² = c².
Missing Side
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Side a
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Side b
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Hypotenuse c
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Triangle Area
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Perimeter
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The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It was named after the ancient Greek mathematician Pythagoras, though the relationship was known to many ancient civilizations. It applies exclusively to right triangles — triangles containing one 90° angle.
Enter any two known side values and leave the third field blank. The calculator will solve for the missing side using the Pythagorean equation. For example, if you know sides a = 3 and b = 4, it will calculate hypotenuse c = 5.
The formula is a² + b² = c², where c is the hypotenuse (the side opposite the right angle) and a and b are the two legs. To solve for a missing side: c = √(a² + b²), a = √(c² − b²), or b = √(c² − a²).
Pythagorean Triples are sets of three positive integers that satisfy a² + b² = c². The most well-known example is (3, 4, 5) because 9 + 16 = 25. Other common triples include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a Pythagorean triple is also a valid triple — for example, (6, 8, 10) is a multiple of (3, 4, 5).
The hypotenuse is the longest side of a right triangle, always located opposite the 90° right angle. In the Pythagorean equation a² + b² = c², the variable c always represents the hypotenuse. It is always longer than either of the two legs (a and b).
Yes. Once all three sides are known, the area of the right triangle is calculated as A = (a × b) / 2, where a and b are the two legs. The legs form the base and height of the triangle, making this formula straightforward to apply.
No — the Pythagorean Theorem only applies to right triangles (triangles with exactly one 90° angle). For non-right triangles, you would use the Law of Cosines: c² = a² + b² − 2ab·cos(C), which is a generalization of the Pythagorean Theorem.
You can use any consistent unit of length — the calculator supports kilometers, meters, centimeters, millimeters, miles, feet, and inches. The important thing is that all sides must be expressed in the same unit. The result will be returned in the same unit you select.