Law of Sines Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The law of sines states a/sin(A) = b/sin(B) = c/sin(C). Knowing one side and two angles, the third angle and the two remaining sides follow directly. For a=10, A=30°, and B=70°, side b is about 18.794 units.
Explanation
The law of sines relates the sides of any triangle to the sines of their opposite angles: the ratio of a side to the sine of the angle facing it is constant throughout the triangle. It is particularly suited to the case where one side and two angles are known (the ASA or AAS case in geometry), the most common situation in triangulation and surveying, where angles are easier to measure than distances. The third angle follows first, simply: the sum of a triangle's three angles is always 180°, so C = 180° − A − B. Once the three angles are known, each missing side is calculated by isolating its variable in the equality of ratios. Unlike the law of cosines, which applies when two sides and the angle between them are known (the SAS case), the law of sines is the formula to use when the starting data is one side and two angles.
Example: a=10, angle A=30°, angle B=70°
Inputs
Side a: 10. Angle A (opposite a): 30°. Angle B: 70°.
Calculation
Angle C = 180° − 30° − 70° = 80°. Side b = 10 × sin(70°) ÷ sin(30°) = 10 × 0.9397 ÷ 0.5 ≈ 18.794. Side c = 10 × sin(80°) ÷ sin(30°) = 10 × 0.9848 ÷ 0.5 ≈ 19.696.
Result
This triangle has an angle C of 80°, a side b of about 18.794 units, and a side c of about 19.696 units.
Frequently asked questions
Why are the angles bounded between 0.1° and 179.8°?
A valid triangle has three strictly positive angles whose sum is exactly 180°: an angle of 0° or 180° would correspond to a completely flattened (degenerate) triangle, with no area. The form bounds exclude these geometrically impossible edge cases, just as the combination of the two entered angles must never reach or exceed 180° for the third angle to stay positive.
When should you use the law of sines rather than the law of cosines?
The law of sines is appropriate when the known data is one side and two angles (ASA or AAS case), or two sides and an angle not between them (SSA case, sometimes with an ambiguity to check). The law of cosines is used instead when two sides and the angle between them are known (SAS case), or the three sides to recover an angle.
Does the law of sines also work for a right triangle?
Yes, the law of sines remains valid for any triangle, including a right triangle: this calculator's second test case (30-60-90 triangle) verifies it directly, recovering exactly the expected hypotenuse. For a specifically right triangle with two known sides, our hypotenuse calculator (Pythagoras) is more direct.