Math
Pythagorean Theorem Calculator
Solve for any missing side of a right triangle using a^2 + b^2 = c^2. Enter any two sides to get the third, plus area and perimeter.
Right triangle sides
Enter any two sides. Leave the side you want to solve for blank. Sides a and b are the legs; c is the hypotenuse (longest side).
Hypotenuse c =
5
Area
6
Perimeter
12
Triangle
Frequently Asked Questions about the Pythagorean Theorem Calculator
What is the Pythagorean theorem formula?
For any right triangle, a^2 + b^2 = c^2, where a and b are the two legs and c is the hypotenuse - the side opposite the right angle. Rearranged to solve for each side: c = sqrt(a^2 + b^2), a = sqrt(c^2 - b^2), b = sqrt(c^2 - a^2). Enter any two sides and the calculator finds the third.
What is the Pythagorean theorem used for in real life?
Construction workers use the 3-4-5 rule to confirm a right angle without a framing square. GPS and mapping software compute straight-line distances using the same formula. Other everyday uses include finding TV or monitor diagonal size from width and height, calculating roof rafter length from run and rise, and checking ladder safety distance from a wall.
What is a 3-4-5 triangle?
A 3-4-5 triangle is the smallest right triangle with all-integer sides: 3^2 + 4^2 = 9 + 16 = 25 = 5^2. Builders use it to square corners - measure 3 ft along one wall, 4 ft along the other, and adjust until the diagonal reads exactly 5 ft. Scaled versions like 6-8-10 or 9-12-15 work equally well.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers (a, b, c) satisfying a^2 + b^2 = c^2. Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Multiplying any triple by the same integer produces another valid triple - for example, doubling (3, 4, 5) gives (6, 8, 10).
Does the Pythagorean theorem work for non-right triangles?
No - it applies only to right triangles. For triangles with any angle, use the Law of Cosines: c^2 = a^2 + b^2 - 2ab x cos(C). When C equals 90 degrees, the cosine term drops to zero and you get the Pythagorean theorem back. Note: the calculator returns no result if the hypotenuse you enter is not longer than each leg, since that input cannot form a valid right triangle.