Sin Triangle Calculator

Background.

Measuring a distance you cannot walk — across a river, between two hilltops — is the oldest job of the law of sines. Lay out one line you can measure, sight the far point from both ends, and the triangle is fixed: one side and two angles determine every other part of it.

The rule behind that is that in any triangle each side is proportional to the sine of the angle facing it: a / sin A = b / sin B = c / sin C, and the shared ratio is the diameter of the circle passing through all three corners. So once you know side a and the angle A opposite it, any other side follows from its own opposite angle, b = a × sin B ÷ sin A. The third angle needs no trigonometry at all, because the angles of a flat triangle add up to 180°.

That is the calculation done here. From side a, its opposite angle A and a second angle B, it returns side b and the third angle C = 180° − A − B. The two angles must leave room for a positive third, so their sum has to stay below 180°.

The pairing matters. Your known side must face one of your known angles. In the river problem the measured baseline lies between the two sighted angles instead, so subtract them from 180° first to get the angle opposite the baseline, and enter that as A.

Two behaviours surprise people. Because sin 60° equals sin 120°, replacing B by its supplement leaves b unchanged even though the triangle changes shape. And the answer gets touchy when A is small: dividing by a small sine magnifies any error in measuring A. The ambiguous case — two sides and an angle not between them, which can give two triangles, one or none — is not handled here.

What is sin triangle calculator?

The law of sines states that the ratio of any side of a triangle to the sine of its opposite angle is the same for all three sides, and equals the diameter of the circle through the triangle's vertices. Solving from one side and two angles — the AAS case, or ASA once the third angle is found — gives exactly one triangle, so the unknown side has a single value.

How to use this calculator.

1. 01Enter the side you know as side a, in any length unit; side b comes out in the same unit.
2. 02Enter angle A, the angle directly opposite that side, in degrees. If your two known angles sit at the two ends of the known side, enter 180° minus their sum as A instead.
3. 03Enter angle B, the angle opposite the side you want to find. A and B together must come to less than 180°.
4. 04Read side b and the third angle C. For the remaining side c, run it again with C as the target angle, keeping the same a and A.

The formula.

b = a · sin B ÷ sin A and C = 180° − A − B. Here a is the known side, A the angle opposite it, B the angle opposite the unknown side b, and C the remaining angle. The first equation is the law of sines rearranged. It follows from dropping a perpendicular from the third corner onto side c: that height can be written both as b · sin A and as a · sin B, and setting the two equal gives the law. Angles are entered in degrees and converted to radians by multiplying by π/180 before the sines are taken. The ratio sin B ÷ sin A sets the scale, so equal angles give equal sides and the larger angle always faces the longer side. When A is small, errors in it are magnified: with a = 5 and B = 60°, A = 5° gives b ≈ 49.68, while A = 6° gives b ≈ 41.43.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.