Mathematics · Geometry
Law of Sines Calculator
Solve any triangle using the Law of Sines given two angles and one side (AAS or ASA).
Calculator
Formula
a, b, c are the side lengths opposite to angles A, B, C respectively. Each side divided by the sine of its opposite angle yields the same constant (the circumdiameter 2R).
Source: Euclidean geometry — standard trigonometric identity for any triangle.
How it works
Given any two angles of a triangle, the third is immediately found since all three must sum to 180°. With one known side and its opposite angle, the common ratio a/sinA is established, and the remaining sides follow by multiplying that ratio by the sine of each remaining angle.
This calculator takes angle A, angle B, and side a (opposite A) as inputs. It computes angle C = 180° − A − B, then applies the sine rule to find sides b and c. It also outputs the circumradius R = a / (2 sin A), the radius of the circle passing through all three vertices.
Worked example
Suppose A = 45°, B = 60°, and a = 10.
Step 1 — Find C: C = 180° − 45° − 60° = 75°.
Step 2 — Establish the ratio: a / sin A = 10 / sin 45° = 10 / 0.7071 ≈ 14.142.
Step 3 — Find b: b = 14.142 × sin 60° = 14.142 × 0.8660 ≈ 12.247.
Step 4 — Find c: c = 14.142 × sin 75° = 14.142 × 0.9659 ≈ 13.660.
Step 5 — Circumradius: R = 14.142 / 2 ≈ 7.071.
Limitations & notes
The Law of Sines cannot directly solve an SSS or SAS triangle — use the Law of Cosines for those cases. The ambiguous case (SSA) arises when given two sides and a non-included angle; this calculator avoids ambiguity by requiring two angles. All inputs must form a valid triangle: both angles must be positive and their sum must be strictly less than 180°, and side a must be positive.
Frequently asked questions
What is the ambiguous case of the Law of Sines?
The ambiguous case (SSA) occurs when you know two sides and a non-included angle, which can produce zero, one, or two valid triangles. This calculator avoids it by requiring two angles instead.
Can I use this calculator for a right triangle?
Yes — simply set one angle to 90° and provide the other angle and the side opposite the first known angle. The sine rule works for all triangle types including right triangles.
What units should I use for side lengths?
Any consistent unit works (metres, feet, cm, etc.) — the output sides will be in the same unit as the input side. Angles must be entered in degrees.
Why is the circumradius computed?
The common sine-rule ratio equals 2R, the diameter of the circumscribed circle, making circumradius a natural by-product. It is useful in geometry proofs and circle-related problems.
What if my two angles already sum to 180° or more?
That configuration is impossible for a real triangle — angle C would be zero or negative. Ensure both angles are positive and their sum is strictly less than 180°.
Last updated: 2025-01-15 · Formula verified against primary sources.