How to use the Law of Sines Calculator
- Enter angle A in degrees.
- Enter side a, the side directly opposite angle A.
- Enter angle B, the second known angle.
- Read angle C, which is whatever remains from 180°.
- Read sides b and c, scaled by the same sine ratio.
How the calculation works
The law of sines states that each side of a triangle divided by the sine of its opposite angle gives the same constant, and that constant equals the diameter of the triangle's circumscribed circle. Knowing any side and its opposite angle therefore fixes the scale of the whole triangle, and every other side follows by multiplying that constant by the appropriate sine.
It is the right tool for the angle-angle-side and angle-side-angle configurations, which is what surveying produces: angles are easy to measure precisely with a theodolite while long distances are not. Triangulating a river crossing or an inaccessible point means measuring one accessible baseline and two angles, then letting the law of sines supply the distances nobody could tape.
One configuration is treacherous. Side-side-angle — two sides and an angle not between them — can produce two different valid triangles, one acute and one obtuse, because the arcsine function returns two candidate angles below 180°. This ambiguous case is a real source of error in navigation and surveying; when a problem gives SSA data, check both solutions against physical plausibility before committing to one.
a/sin A = b/sin B = c/sin C = 2R, where R is the circumradiusSource: Law of sines, formalised by Nasir al-Din al-Tusi, Treatise on the Quadrilateral (13th century).
Worked example
Surveying across a river: a 7 m baseline subtends 35° at one end and 65° at the other.
- A = 35°, a = 7 m (the side opposite the 35° angle), B = 65°.
- Third angle: C = 180 − 35 − 65 = 80°.
- Common ratio: 7 / sin(35°) = 7 / 0.5736 = 12.204.
- Side b = 12.204 × sin(65°) = 12.204 × 0.9063 = 11.06 m.
Side b is 11.06 m and side c is 12.02 m — the crossing distance obtained without entering the water.
Frequently asked questions
When should I use the law of cosines instead?+
When you know three sides, or two sides and the angle between them. The law of sines needs a matched side-and-opposite-angle pair to establish scale.
What is the ambiguous case?+
Two sides plus a non-included angle can describe two distinct triangles. Always check whether both are geometrically possible before choosing.
Why does the calculator show a circumdiameter?+
The shared ratio is exactly the diameter of the circle through all three vertices, which is useful in circle-fitting and mechanism design.
Do angles have to be in degrees?+
Enter degrees here; the calculator converts to radians internally for the trigonometric functions.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.