Right Triangle Calculator
Solve a right triangle from any two known values — two legs, a leg and the hypotenuse, or a side and an acute angle — for missing sides, angles, and trig ratios.
How It's Calculated
Formula
a^2 + b^2 = c^2 \qquad \sin A = \frac{a}{c} \qquad \cos A = \frac{b}{c} \qquad \tan A = \frac{a}{b} \qquad A + B = 90°A right triangle has one 90-degree angle, so its two remaining angles (A and B) are always acute and always sum to exactly 90 degrees. Knowing any two values — two sides, or one side and one acute angle — is enough to solve for everything else: the Pythagorean theorem (a² + b² = c²) relates the three sides, and the basic trig ratios (sin A = a/c, cos A = b/c, tan A = a/b) relate a side pair to angle A. This calculator supports four known-value combinations: two legs, a leg plus the hypotenuse, a leg plus its opposite angle, or the hypotenuse plus angle A. Angle A is always the acute angle opposite side a (the leg referenced by whichever mode you choose); angle B is its complement.
Worked Examples
Two legs: a = 3, b = 4 (the classic 3-4-5 triangle)
- Hypotenuse: c = √(a² + b²) = √(9 + 16) = √25 = 5
- Angle A: tan⁻¹(a/b) = tan⁻¹(3/4) ≈ 36.87°
- Angle B: 90 − 36.87 ≈ 53.13°
- sin A = 3/5 = 0.6, cos A = 4/5 = 0.8, tan A = 3/4 = 0.75
Leg + angle: a = 3, angle A ≈ 36.87°
- Hypotenuse: c = a / sin(A) = 3 / sin(36.87°) ≈ 5
- Other leg: b = a / tan(A) = 3 / tan(36.87°) ≈ 4
- Angle B: 90 − 36.87 ≈ 53.13°
Frequently Asked Questions
How is this different from the Pythagorean Theorem Calculator?
The Pythagorean Theorem Calculator solves for exactly one missing side from two known sides — no angles involved. This Right Triangle Calculator additionally accepts a side plus an acute angle as your known values, and always reports both acute angles and the sin/cos/tan trig ratios alongside the missing sides.
How is this different from the Triangle Calculator?
The Triangle Calculator works with any three given side lengths for any triangle (not necessarily a right triangle), checking validity and computing area and side classification — it never solves for a missing value or an angle. This calculator always assumes a right angle and solves for whatever you didn't provide.
What does 'Angle A' mean in every mode?
Angle A is always the acute angle opposite the side labeled 'leg a' in whichever mode you selected — check each field's helper text for exactly which known value plays that role in the current mode. Angle B is simply 90° minus angle A.
Why does the calculator reject some leg + hypotenuse combinations?
The hypotenuse is always the longest side of a right triangle. If the hypotenuse value you enter isn't strictly greater than the known leg, those two numbers can't describe a real right triangle, so the calculator reports that clearly instead of taking the square root of a negative number.
Why must the angle be strictly between 0 and 90 degrees?
In any right triangle, the two non-right angles are always acute — each strictly between 0° and 90°. An angle of exactly 0° or 90° (or beyond) can't be one of those two angles, so it isn't a valid input for this calculator's angle-based modes.