Two legs, a leg and the hypotenuse, or a leg and an angle — each is enough to determine everything else.
Legs 3 and 4 is the calculator's starting example, and it resolves to the best-known right triangle there is: hypotenuse c = √(3² + 4²) = √25 = 5. The two acute angles are 36.87° and 53.13° (they must add to 90°), the area is ½ × 3 × 4 = 6, and the perimeter is 3 + 4 + 5 = 12.
3-4-5 is the smallest set of whole numbers that satisfies a² + b² = c² with no rounding at all — which is exactly why it is the default here and on the Coordinate Distance Calculator, whose own default points, (0,0) to (3,4), are the same triangle viewed as two coordinates instead of two legs. Multiples of it (6-8-10, 9-12-15, and so on) are just as exact, which is why builders use a 3-4-5 tape-measure trick to square a corner without a protractor.
It only solves right triangles, and needs one side length no matter which mode is chosen — two angles alone leave the size undetermined, as the FAQ below explains. For a triangle without a right angle, or one where all that is known is three angles or a mix of arbitrary sides, use the Triangle Calculator's SSS/SAS/ASA solvers instead; for the narrower case of finding one missing side once a right angle is already known, the Pythagorean Theorem Calculator is the more direct tool.
A right triangle already has one angle fixed at 90°, so any two further pieces of information determine it completely. Two sides, or one side and one acute angle, and everything else follows.
What does not work is two angles. Knowing both acute angles fixes the shape but not the size — you would have a family of similar triangles, all the same proportions at different scales. At least one length is always required.
With an angle A and a side, the three ratios do the work:
Given leg a = 3 and angle A = 30°, the hypotenuse is a ÷ sin(30°) = 6 and the other leg is a ÷ tan(30°) = 5.1962.
The angles of any triangle add to 180°, and one of them here is already 90°, so the other two must share the remaining 90° between them. Find one and you have the other by subtraction — no trigonometry needed for the second.
The area is simply ½ × a × b, using the two legs. They are perpendicular to each other, so one serves as the base and the other as the height with no extra work — the only triangle where that is true without construction.
No. The two acute angles determine the shape but not the size, giving infinitely many similar triangles. You need at least one side length.
Because the triangle already contains a 90° angle and the three must total 180°. Both remaining angles have to be acute, and they always sum to exactly 90°.
It depends on the angle you are working from. The opposite leg is the one not touching that angle; the adjacent leg is the one that does, excluding the hypotenuse. Swap the angle and the two swap roles.
In a right triangle you already do. The two legs are perpendicular, so area is ½ × leg a × leg b with no separate height needed.
If you know two sides and just want the third, the Pythagorean Theorem Calculator is the quicker route. For triangles without a right angle use the Triangle Calculator, and for the trigonometric functions directly, the Scientific Calculator.