Pythagorean theorem calculator — find the hypotenuse or a missing leg

How to use the Pythagorean calculator

1. Under Solve for, choose Hypotenuse c, Leg a or Leg b.
2. Fill in the two boxes that remain visible. Both must be greater than zero.
3. Read the Result, then the Area, Perimeter and the two acute angles.
4. Follow the Working box line by line, and use the diagram to check which side is which.

What it does and when to use it

In a right triangle the two short sides are called legs and the long side is the hypotenuse. If you know any two, the third is fixed. That is handy for the length of a ladder against a wall, the diagonal of a screen or a room, the length of a roof rafter, or the straight-line distance across a field. In homework it is the first tool for “find x” questions on right triangles.

For triangles with no right angle, use the triangle solver. The same theorem gives the distance between two points on a graph, and the trigonometry calculator helps when you know an angle instead of a second side.

How it works

The theorem says a² + b² = c². Rearranging gives the three cases the tool uses:

• Hypotenuse: c = √(a² + b²)
• Leg a: a = √(c² − b²)
• Leg b: b = √(c² − a²)

Once all three sides are known, the tool finds area = a × b ÷ 2 (the legs are base and height), perimeter = a + b + c, and each acute angle from the arctangent of the two legs.

Worked examples

Hypotenuse, step by step. Legs 6 and 8.

1. Square each leg: 6² = 36 and 8² = 64.
2. Add the squares: 36 + 64 = 100. This is c².
3. Take the square root: √100 = 10.

The working ends with c = √100 = 10. The tool also gives area 24 and perimeter 24.

Missing leg. Leg a = 9 and hypotenuse 15. This time you subtract: b² = 15² − 9² = 225 − 81 = 144, so b = 12. The angles are 36.869898° and 53.130102°.

An answer that is not whole. Legs 5 and 7 give c² = 74 and Result: c = 8.602325267. Most right triangles look like this. Whole-number sets such as 3-4-5 and 5-12-13 are special cases called Pythagorean triples.

Limits and tips

• Keep all sides in the same unit. The tool has no unit menu, so the answer is in whatever unit you used.
• Results show up to 10 significant figures. Round at the end, not in the middle, to avoid small errors.
• A side of 0 or a blank box gives a prompt instead of a result.
• The calculation runs on your device.