Pythagorean Theorem Calculator
How do I find the hypotenuse?
Result
How it is calculated
The hypotenuse comes from the Pythagorean theorem: c = √(a² + b²), where a and b are the two legs that meet at the right angle, in any one unit of length. The tool also returns the area, a × b / 2, and the perimeter, a + b + c. Area comes out in the square of your unit and the other two in the unit itself.
Both legs must be greater than zero; zero or negative values give no result. Answers are rounded to two decimals for display only, so a hypotenuse like √2 is shown as 1.41 although the true value continues. It only works when you know the legs, not when one of them is the missing side.
Worked example
Take legs a = 5 and b = 12. First square them: 25 and 144, which add up to 169. The square root of 169 is the hypotenuse, 13.00. The area is 5 × 12 / 2 = 30.00, and the perimeter is 5 + 12 + 13 = 30.00. Because 5, 12 and 13 are whole numbers, this is a Pythagorean triple and the result has no rounding.
Common mistakes
- Entering the hypotenuse as one of the legs. The tool assumes both inputs are the shorter sides next to the right angle, so the result will be too large.
- Mixing units, such as one leg in centimetres and the other in metres.
- Applying it to a triangle that has no right angle; the theorem does not hold there.
- Reading the area as a length: it is in square units, while the hypotenuse and perimeter are in plain units.
About this calculator
This tool applies the Pythagorean theorem (a² + b² = c²) to a right triangle's two legs to find the length of the hypotenuse, the side opposite the right angle, and then also reports the triangle's area and perimeter using those three side lengths. The relationship holds only for right triangles, so the calculator assumes a 90-degree angle between the two given sides.
It is used by students checking geometry and trigonometry homework, and in a surprising number of practical situations: carpenters and builders squaring a corner or cutting a diagonal brace, electricians and installers figuring out the shortest cable run between two points at different heights, and anyone measuring the diagonal of a screen, room or piece of furniture from its width and height. Because the theorem shows up anywhere two perpendicular measurements need to be turned into a straight-line distance, it's one of the most reused formulas outside the classroom.
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