Right Triangle Side and Angle Calculator

SOH-CAH-TOA is the mnemonic for the three basic trig ratios in a right triangle. SOH: Sine = Opposite / Hypotenuse. CAH: Cosine = Adjacent / Hypotenuse. TOA: Tangent = Opposite / Adjacent. "Opposite" and "adjacent" are always read relative to the acute angle you are working with, never the right angle. With angle A as the reference, leg a (opposite A) is the opposite side and leg b is the adjacent side, so sin(A) = a / c, cos(A) = b / c, and tan(A) = a / b. This calculator uses those three identities together with B = 90 - A to recover the missing values from any compatible pair.

For any right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2. The hypotenuse is always the longest side because squaring is monotonic on positive numbers, so c = sqrt(a^2 + b^2) is strictly larger than either leg. Rearranging gives the two leg formulas: a = sqrt(c^2 - b^2) and b = sqrt(c^2 - a^2). When you pick "two legs" or "leg and hypotenuse", the calculator uses exactly these formulas first to find the missing side, then derives the angles from the resulting ratios.

All sixteen primitive triples with hypotenuse up to 97, including the classics: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29, and 12-35-37. Scaled multiples like 6-8-10 or 9-12-15 are also detected, because the sides are still positive integers that satisfy a^2 + b^2 = c^2. If your triangle is not an integer triple, the calculator still surfaces the closest primitive shape by side-ratio so you can see how near you are to, for example, a 3-4-5 or 5-12-13 form.

Degrees split a full circle into 360 equal parts, so a right angle is 90 degrees. Radians measure the same angle as the arc length on a unit circle, so a full circle is 2 x pi radians and a right angle is pi / 2 radians (about 1.5708). The conversion is radians = degrees x pi / 180. Trig functions in nearly every programming language, including the Math.sin and Math.cos calls behind this calculator, take radians as input. The calculator reports both units side by side so you can plug the result into spreadsheets, CAD tools, or code without converting by hand.

The inradius r is the radius of the largest circle that fits inside the triangle and touches all three sides. The circumradius R is the radius of the unique circle that passes through all three vertices. For a right triangle, both have clean closed forms: r = (a + b - c) / 2 and R = c / 2 (the right-angle vertex always sits on the circumscribed circle, with the hypotenuse as the diameter). These show up in geometry homework, in machining when you need a fillet that touches every face of a right-angled bracket, and in CAD when you want the bounding circle for a triangular cut.