Law of Sines Calculator
Enter a side, the angle opposite it, and a second angle to find the side opposite that angle — the law of sines does the rest.
One side, two angles
Give the calculator a known side a, its opposite angle A, and the angle B opposite the side you want, and it returns b = a × sin(B) / sin(A).
Angles in degrees
Enter both angles in degrees, each below 180°, with the two angles summing to less than 180° so a real triangle remains.
What does the law of sines do?
Sides and their opposite angles
The law of sines calculator finds an unknown side of a triangle from one known side, the angle opposite it, and a second angle. The law of sines says that in any triangle the ratio of a side to the sine of its opposite angle is the same for all three sides, so a / sin(A) = b / sin(B). Rearranging gives the side opposite angle B directly: b = a × sin(B) / sin(A). It is the tool behind surveying distances you cannot measure, navigation fixes, and any geometry problem where you know an angle and the side facing it.
Enter side a, angle A opposite it, and angle B opposite the side you want to get b = a × sin(B) / sin(A) instantly.
The side opposite angle B equals the known side a, scaled by the ratio of the sine of angle B to the sine of angle A.
b = a × sin(B) / sin(A)Both angles go in as degrees and are converted to radians before the sine is taken. Because the calculator starts from one side and two angles, the third angle (and therefore the whole triangle) is fixed, so the answer is always a single, unambiguous side.
Suppose a triangle has side a = 10 opposite angle A = 30°, and you want the side opposite angle B = 45°.
Take the sine of each angle
sin(45°) ≈ 0.70711 and sin(30°) = 0.5 — the two sines in the ratio.
Form the ratio
sin(45°) / sin(30°) ≈ 0.70711 / 0.5 ≈ 1.41421 — how much bigger b is than a.
Scale the known side
10 × 1.41421 ≈ 14.142136 — the side opposite angle B.
The result is the length of the side facing angle B, expressed in the same unit you used for side a — if a was in metres, b is in metres. The ratio sin(B) / sin(A) tells the whole story: when angle B is larger than angle A, its opposite side is longer (b > a), and when B is smaller, the side is shorter. In the example, B = 45° is larger than A = 30°, so b ≈ 14.14 comes out longer than a = 10. This mirrors a basic fact about triangles — the largest angle always faces the longest side — and the law of sines makes that relationship exact. Once you have side b, the same ratio lets you solve for the remaining side or angle and pin down the entire triangle.
The formula is exact, but a few conditions keep it valid.
Degrees, two angles, and the ambiguous SSA case
Enter both angles in degrees — each greater than 0° and less than 180°, with the two together below 180° so a genuine triangle exists. This calculator takes one side and two angles, which always defines a single triangle. The classic pitfall is the ambiguous SSA case — two sides and a non-included angle — where the law of sines can yield zero, one, or two triangles; working from two angles, as here, sidesteps that ambiguity entirely.