A right triangle is fully determined by any two of its sides, because the fixed 90° angle removes a degree of freedom that a general triangle does not have. Pythagoras’ theorem gives the third side directly, and the inverse trigonometric functions then give both remaining angles from the side lengths alone — no additional measurement needed.
Which ratio to reach for
The mnemonic SOH-CAH-TOA encodes the three ratios: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. When you know two sides and want an angle, the practical approach is to pick whichever ratio uses exactly the two sides you already have, then apply its inverse function. Knowing the opposite and hypotenuse, for instance, points directly at arcsine rather than arccosine or arctangent — there is no need to solve for the third side first if the angle is all you actually want.
The two acute angles always sum to 90°
This follows from the fact that every triangle’s interior angles sum to 180°, and a right triangle has already spent 90° of that budget on the right angle itself. The two remaining angles are called complementary, and this relationship is exactly why sin A = cos B for the two acute angles in the same triangle — it is also the historical origin of the word “cosine,” which literally means the sine of the complementary angle.
Pythagorean triples
Most right triangles have at least one irrational side length, but a special family — Pythagorean triples — have all three sides as whole numbers. The smallest and most famous is 3-4-5; others include 5-12-13, 8-15-17, and 7-24-25. Any whole-number multiple of a triple is itself a valid triple (6-8-10 is just 3-4-5 doubled), which is why builders and carpenters use the 3-4-5 method — measuring 3 units along one edge, 4 along the perpendicular edge, and checking the diagonal comes out to exactly 5 — to verify a corner is truly square using nothing more than a tape measure.
The inradius, and why it is worth knowing
Every triangle has an inscribed circle — the largest circle that fits entirely inside it, touching all three sides. For a right triangle specifically, the radius of that circle has an unusually clean formula: r = (a + b − c) / 2, using the two legs and the hypotenuse. This shortcut does not exist in anywhere near as simple a form for a general (non-right) triangle, where the inradius instead requires knowing the full area and perimeter through a more general formula. It occasionally shows up in construction and packing problems where the largest circular object that fits in a right-angled corner needs to be sized quickly.
What two sides cannot tell you
Given only two sides, this calculator assumes they are the two legs (the sides adjacent to the right angle), not one leg and the hypotenuse. If you have a leg and the hypotenuse instead, the third side is found by rearranging Pythagoras (b = √(c² − a²)) rather than the direct addition used here — worth checking which two measurements you actually have before reading off a result.