What is Pythagoras's theorem?

Pythagoras's theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a squared plus b squared equals c squared. You can rearrange this to find any side if you know the other two.

How do you find a missing leg of a right triangle?

If you know the hypotenuse c and one leg a, you can find the missing leg b by rearranging Pythagoras's theorem: b = square root of (c squared minus a squared). Make sure c is larger than a, otherwise the value under the square root is negative and no real solution exists.

What is the difference between the hypotenuse and a leg?

In a right-angled triangle, the hypotenuse is the longest side and is always opposite the right angle (90 degrees). The other two shorter sides are called the legs or catheti. The hypotenuse is always strictly longer than either leg.

Hypotenuse Calculator

Pythagoras's theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a squared plus b squared equals c squared, where c is the hypotenuse. This relationship lets you find any one side when you know the other two, and this calculator handles all three versions of the problem. If you know both legs a and b, it calculates the hypotenuse c = the square root of (a squared plus b squared). If you know the hypotenuse c and one leg a, it finds the missing leg b = the square root of (c squared minus a squared). The same works if you know c and b and want a. To use the calculator, enter two known side lengths and leave the third field blank. The calculator automatically detects which two you have provided, solves for the missing one, and also returns the triangle's perimeter, its area (half the base times height for a right triangle), and the two non-right angles in degrees. The default example uses a=3 and b=4, which gives the hypotenuse c=5, one of the most famous results in all of mathematics, the 3-4-5 Pythagorean triple. This tool is useful for students at every level, builders checking whether a corner is truly square (a 3-4-5 check), surveyors calculating distances, and anyone who needs the third side of a right triangle. Results are for reference only.

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Leg a (leave blank to solve)

Leg b (leave blank to solve)

Hypotenuse c (leave blank to solve)

5.00

hypotenuse (c)

Perimeter12.00

Area6.00

Angle A36.87°

Angle B53.13°

How it works

The calculator reads all three fields. If a and b are given and c is blank, it computes c = √(a² + b²). If c and a are given, b = √(c² − a²). If c and b are given, a = √(c² − b²). If all three are provided, it checks whether they satisfy a² + b² = c² and reports the result. Area = ½ × a × b. Angle A = arcsin(a/c), Angle B = arcsin(b/c), both in degrees.

Worked example

With leg a=3 and leg b=4 (the default values), the hypotenuse c = √(9 + 16) = √25 = 5.00. Perimeter = 3 + 4 + 5 = 12.00. Area = ½ × 3 × 4 = 6.00 square units. Angle A = arcsin(3/5) = 36.87°, Angle B = arcsin(4/5) = 53.13°. This is the 3-4-5 right triangle.

Related calculators

Triangle Calculator: solve any triangle, not just right triangles, from sides and angles.
Distance Formula Calculator: apply Pythagoras to find the distance between two coordinate points.
Heron's Formula Calculator: find triangle area from three sides without needing the height.
Area Calculator: find the area of other 2D shapes including circles and rectangles.

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