Law of Sines Calculator
The remaining sides of a triangle from two angles and one side — the law of sines — with the third angle, the area, the perimeter and the circumscribed circle's diameter, which the ratio in the law equals.
In any triangle each side divided by the sine of the angle opposite it gives the same number — and that number is the diameter of the circle through the three corners.
How the law of sines calculator works
In any triangle each side divided by the sine of the angle opposite it gives the same number — and that number is the diameter of the circle through the three corners. Given two angles, the third is whatever brings the sum to 180°; given any one side, the common ratio is fixed and every other side follows. It is the tool for surveying and navigation, where angles are easy to measure and distances are not.
Formula: a / sin A = b / sin B = c / sin C = 2R; C = 180° − A − B
Worked examples
| Inputs | Side b, opposite angle B | Note |
|---|---|---|
| 42° and 71° with a = 9 | 12.7175 | b about 12.7 |
| Equilateral | 5 | every side 5 |
| A right triangle | 1.73205 | the 1 : √3 : 2 triangle |
FAQFrequently asked questions
What is the law of sines?
In any triangle, a side divided by the sine of its opposite angle is the same for all three sides — and equals the diameter of the circumscribed circle.
When do I use it?
When you know two angles and a side (AAS or ASA). Two sides and a non-included angle also uses it, but that case can have two solutions; this page takes angles.
Why does the third angle come for free?
Because the angles of a triangle sum to 180°. Two angles fix the shape completely; the side only fixes the size.
What is the circumscribed circle?
The circle passing through all three corners. Its diameter is the law-of-sines ratio, which is why the ratio has a geometric meaning and not just a numerical one.
How do surveyors use it?
Measure a baseline and the angles to a distant point from each end; the law of sines gives the distances without walking there. Triangulation is this, repeated.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.