Right Triangle Calculator
Hypotenuse: 0
| Step | Calculation |
|---|
How a Right Triangle Is Solved
A right triangle has one 90° angle, and its sides follow the Pythagorean theorem, a² + b² = c², where c is the hypotenuse (the side opposite the right angle) and a and b are the two legs. Once any two of the three sides are known, the third follows directly from that equation. When only one side and one non-right angle are known, the trigonometric ratios sine, cosine, and tangent fill in the rest: sin(A) = a/c, cos(A) = b/c, and tan(A) = a/b, where A is the angle opposite side a. Because the two non-right angles always sum to 90°, finding one gives you the other for free (B = 90° − A).
Area, Perimeter, and the Altitude to the Hypotenuse
Area is simply half the product of the two legs, Area = ½ab, since the legs themselves form a right angle and act as base and height. Perimeter is the sum of all three sides, a + b + c. A less obvious but useful figure is the altitude drawn from the right angle down to the hypotenuse: h = ab / c. This comes from the fact that the triangle's area can also be written as ½ × c × h, so setting the two area expressions equal and solving for h gives that relationship.
Special Cases and Related Tools
Certain right triangles recur constantly in practice: the 3-4-5 triangle and its multiples, and the 45-45-90 and 30-60-90 "special" triangles where side ratios are fixed exactly (1:1:√2 and 1:√3:2 respectively). If you only need to solve for a single missing side from the other two, the Pythagorean theorem calculator is a more focused tool. For triangles that aren't right triangles — where you'd instead need the Law of Sines or Law of Cosines — use the general triangle calculator.
Frequently Asked Questions
What two values do I need to solve a right triangle?
Any two of the following, as long as at least one is a side length: both legs, one leg plus the hypotenuse, one leg plus a non-right angle, or the hypotenuse plus a non-right angle. Knowing only the two non-right angles (which always sum to 90 degrees) isn't enough, since that fixes the triangle's shape but not its size.
How is the height (altitude) to the hypotenuse calculated?
It equals the product of the two legs divided by the hypotenuse: h = (a x b) / c. This comes from equating the two ways of expressing the triangle's area - one using the legs as base and height, and one using the hypotenuse as the base - since a right triangle's area is unambiguous regardless of which side you use as the base.