Right Triangle Calculator
Enter the two short sides of a right triangle for the hypotenuse, both acute angles, the area and the perimeter — Pythagoras and the trigonometry in one place.
In a right triangle the two short sides (the legs) meet at 90°, and the hypotenuse opposite that angle follows from Pythagoras: c = √(a² + b²).
How the right triangle calculator works
In a right triangle the two short sides (the legs) meet at 90°, and the hypotenuse opposite that angle follows from Pythagoras: c = √(a² + b²). The acute angles come from the ratios of the sides — tan A = a ÷ b — and always add to 90°, since the right angle has already used half of the 180°. The area is simply half the product of the legs, because each leg is the height to the other.
Formula: a² = b² + c² − 2bc·cos A; a / sin A = b / sin B = c / sin C; area = ½ab·sin C
Worked examples
| Inputs | Area | Note |
|---|---|---|
| Legs 3 and 4 | 6 | hypotenuse 5, area 6 |
| Legs 6 and 8 | 24 | hypotenuse 10 — a scaled 3-4-5 |
| Equal legs of 5 | 12.5 | a 45-45-90 triangle |
FAQFrequently asked questions
What is the Pythagorean theorem?
In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
How do I find the angles of a right triangle?
From the side ratios: tan A = opposite ÷ adjacent, so A = arctan(a ÷ b). The other acute angle is 90° minus that.
What are the common right triangles?
3-4-5 and its multiples, 5-12-13, 8-15-17, and the two special ones — 45-45-90 and 30-60-90.
What is the area of a right triangle?
Half the product of the two legs, since each leg is the height measured to the other.
How do I know if a triangle is right-angled?
Check whether the two shorter sides squared add up to the longest squared. The SSS page classifies it for you.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.