Hypotenuse Calculator

What this calculator does

This finds the hypotenuse of a right triangle, the longest side, opposite the right angle. You can find it from the two other sides, or from one side together with an angle.

Pick the method that matches what you know, and the tool returns the hypotenuse.

Using the calculator

1. Choose the method: 2 sides, 1 side and an adjacent angle, or 1 side and an opposite angle.
2. Enter the values. Angles fall between 0 and 90°, in degrees or radians.
3. Press Calculate.

The result is the length of the hypotenuse.

From two sides: Pythagoras

If you know both of the shorter sides, the hypotenuse comes straight from the Pythagorean theorem: c = √(side1² + side2²). You square the two sides, add them, and take the square root. Nothing more than the theorem at work.

From a side and an angle: trig

If you have just one side and an angle, trigonometry gets you the rest of the way. When the side you know is adjacent to the angle, the hypotenuse is side ÷ cos(angle), because the cosine is adjacent over hypotenuse. When the side is opposite the angle, it is side ÷ sin(angle), since the sine is opposite over hypotenuse. The tool applies whichever fits the choice you made.

Two worked examples

Two sides: with sides of 3 and 4, the hypotenuse is √(3² + 4²) = √25 = 5.

Side and angle: with a side of 5 adjacent to a 60° angle, the hypotenuse is 5 ÷ cos(60°) = 5 ÷ 0.5 = 10.

This tool, or the Pythagorean theorem calculator

The Pythagorean theorem calculator finds any side of a right triangle, either leg or the hypotenuse, plus the area, but always from two sides. This tool finds specifically the hypotenuse, yet from two sides or from a single side and an angle. So reach for this one when the hypotenuse is what you are after, especially if you have an angle rather than both legs. For angles as well as sides, the trigonometry calculator solves the whole triangle.

Questions people ask

How do you find the hypotenuse from two sides?

Square both sides, add them, and take the square root: c = √(side1² + side2²).

How do you find it from a side and an angle?

Divide the side by the cosine of the angle if the side is adjacent, or by the sine if it is opposite.

What is the hypotenuse of sides 3 and 4?

5, since √(9 + 16) = √25 = 5.

How is this different from the Pythagorean theorem calculator?

That one finds any side from two sides. This one finds only the hypotenuse, but also from a side and an angle.

References

A note on the idea behind it. The hypotenuse follows from the Pythagorean theorem when two sides are known, c = √(side1² + side2²). From one side and an angle, it comes from the cosine or sine ratio, dividing the side by the cosine of an adjacent angle or the sine of an opposite angle. For further reading, see Hypotenuse.

1. The hypotenuse, found from two sides by the Pythagorean theorem, or from a side and an angle through the cosine or sine ratio.