How to use the calculator
Type the two sides you know and leave the third box empty. Press Calculate and the missing side appears, along with the area, the perimeter and the two acute angles in degrees. The details panel shows each substitution so you can follow or check the working.
- Leg a and leg b are the two sides that meet at the right angle.
- Hypotenuse c is the side opposite the right angle. It is always the longest side.
Use the same length unit for every side. The result is in that unit, and the area is in that unit squared.
The formula
In every right triangle the square of the hypotenuse equals the sum of the squares of the two legs:
a² + b² = c²
Rearranged, it gives each side:
- Hypotenuse:
c = √(a² + b²) - Missing leg:
a = √(c² − b²)orb = √(c² − a²)
The acute angles come from basic trigonometry: angle A, opposite leg a, is arctan(a / b), and angle B is 90° − A. The area is half the product of the legs, a × b / 2.
Worked example: finding the hypotenuse
Legs of 6 and 8:
- c² = 6² + 8² = 36 + 64 = 100
- c = √100 = 10
- Area = 6 × 8 / 2 = 24, perimeter = 6 + 8 + 10 = 24
- Angle A = arctan(6 / 8) ≈ 36.87°, angle B ≈ 53.13°
Worked example: finding a leg
A 5 m ladder leans against a wall with its foot 1.5 m from the wall. How high does it reach? The ladder is the hypotenuse:
- a² = 5² − 1.5² = 25 − 2.25 = 22.75
- a = √22.75 ≈ 4.7697 m
The ladder reaches about 4.77 m up the wall, at an angle of about 72.54° to the ground.
Good to know
- Whole-number solutions such as 3-4-5, 5-12-13 and 8-15-17 are called Pythagorean triples. Any multiple of a triple, like 6-8-10, is also one.
- The hypotenuse must be longer than either leg. If it is not, no right triangle exists and the calculator tells you so.
- Results are shown with up to 10 significant digits. Square roots of most numbers are irrational, so the last digit is rounded.
- The theorem works only for right triangles. For other triangles, use the law of cosines, which our triangle calculator applies for you.