Calculate the hypotenuse or missing side of a right triangle using the Pythagorean theorem (a² + b² = c²). Step-by-step solutions with area, angles, and altitude.
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Use this Pythagorean theorem calculator to find the hypotenuse or any missing side of a right triangle. Enter two known sides and instantly get the third side, plus area, perimeter, angles, altitude to hypotenuse, and median. Includes step-by-step solutions and Pythagorean triple detection.
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. The hypotenuse is the longest side, always opposite the 90-degree angle. This relationship, discovered by Greek mathematician Pythagoras around 500 BCE, is written as a² + b² = c², where c is the hypotenuse and a, b are the legs.
Hypotenuse Formula
c = sqrt(a² + b²)Find the hypotenuse or any missing side of a right triangle instantly without manual calculation errors.
See every calculation step with LaTeX-rendered formulas so you learn the process, not just the answer.
Get all sides, angles, area, perimeter, altitude to hypotenuse, and median in one calculation.
Automatically detects Pythagorean triples (3-4-5, 5-12-13) and special triangles (45-45-90, 30-60-90).
Load common examples like building corners, TV screen diagonals, and roof pitch measurements with one click.
Use the 3-4-5 rule to verify corners are perfectly square. Measure 3 ft along one wall, 4 ft along the other — the diagonal should be exactly 5 ft.
Calculate the diagonal screen size from width and height. A 16:9 TV with 16-inch width and 9-inch height has a diagonal of 18.36 inches.
Find rafter length from rise and run measurements. A roof with 5-foot rise and 12-foot run needs a 13-foot rafter (5-12-13 triple).
Calculate straight-line distance between two points on a grid instead of traveling along perpendicular paths.
Determine safe ladder placement by calculating the proper base distance from a wall given the ladder length and desired height.
Square both legs (a² and b²), add them together, then take the square root: c = sqrt(a² + b²). For example, with legs 3 and 4: c = sqrt(9 + 16) = sqrt(25) = 5. This calculator does it instantly and shows every step.
The hypotenuse formula comes from the Pythagorean theorem: c = sqrt(a² + b²), where a and b are the two legs (shorter sides) and c is the hypotenuse (longest side opposite the right angle).
No. The hypotenuse is always the longest side of a right triangle because it is opposite the largest angle (90 degrees). If your calculation gives a shorter hypotenuse, check that your inputs form a valid right triangle.
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². Common triples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Multiples of these also work, like (6, 8, 10). This calculator automatically detects triples.
No, it only works for right triangles (those with exactly one 90-degree angle). For other triangles, use the Law of Cosines: c² = a² + b² - 2ab * cos(C). We also have a Law of Sines calculator for oblique triangles.
In a 45-45-90 triangle, both legs are equal. The hypotenuse is always leg * sqrt(2), approximately 1.414 times the leg length. For example, if each leg is 5, the hypotenuse is 5 * sqrt(2) = 7.071.
In a 30-60-90 triangle, the hypotenuse is always twice the shortest leg (the one opposite the 30-degree angle). The longer leg is the shortest leg times sqrt(3). For example, if the short leg is 4, the hypotenuse is 8.
The altitude to the hypotenuse is a line drawn from the right-angle vertex perpendicular to the hypotenuse. Its length is h = (a * b) / c, where a and b are the legs and c is the hypotenuse. It divides the original triangle into two smaller similar triangles.
Builders use the 3-4-5 method to square corners. TV sizes are the hypotenuse of width and height. Roofers calculate rafter lengths from rise and run. Surveyors find distances. Navigation uses it for shortest paths. Even GPS and video games rely on it.
Rearrange the formula: a = sqrt(c² - b²). Enter the hypotenuse and the known leg, then select which leg to solve for. The calculator handles this automatically and shows the step-by-step derivation.