The law of sines links each side of a triangle to the angle across from it. Know one matched pair (a side and its opposite angle) plus one more side or angle, and it gives you everything else. This calculator covers the three cases where that works: two angles and a non-included side (AAS), two angles and the included side (ASA), and two sides with a non-included angle (SSA), including the ambiguous case where two triangles fit.
How to use the law of sines calculator
- In What do you know?, choose AAS, ASA or SSA.
- Enter Angle A, then Angle B and Side a (AAS), Angle B and Side c (ASA), or Side a and Side b (SSA). Side a is opposite angle A, side b opposite angle B, and side c lies between them.
- Pick the Units for the sides (meters by default) and the Angle unit, degrees or radians.
- For a single triangle, the headline is the area. The tape lists every side and angle (solved ones highlighted), the perimeter, the triangle type, the common ratio a ÷ sin A and the circumradius, plus a table of heights, medians and angle bisectors. When SSA allows two triangles, both appear side by side with a diagram each.
The law of sines formula
To find a side, multiply the common ratio by the sine of the opposite angle: b = (a ÷ sin A) × sin B. To find an angle, divide and take the inverse sine: sin B = b sin A ÷ a.
Why the ratio is the circumcircle’s diameter
Draw the circle through the three corners and the diameter from B to a point D. The inscribed angle at D stands on the same arc BC as angle A, so it equals A, and angle BCD is a right angle because BD is a diameter. So sin A = BC ÷ BD = a ÷ 2R, and a ÷ sin A = 2R. The same argument works for every side. (If A is obtuse, the angle at D is 180° − A, which has the same sine.)
Worked example: measuring across a river
A survey crew needs the width of a river without crossing it. They set stakes A and B 120 m apart along one bank and sight a tree on the far bank from each. The angle between the baseline and the tree is 62° at A and 71° at B. Choose ASA, enter Angle A = 62, Angle B = 71 and Side c = 120, with units in m.
- Angle at the tree: C = 180° − 62° − 71° = 47°.
- Common ratio: 120 ÷ sin 47° ≈ 164.08 m.
- Stake A to the tree (side b): 164.08 × sin 71° ≈ 155.14 m.
- Stake B to the tree (side a): 164.08 × sin 62° ≈ 144.87 m.
- River width, listed as the height to side c: 155.14 × sin 62° ≈ 136.98 m.
This is triangulation: one taped baseline plus angles, which are easy to measure precisely, gives distances you cannot reach. The Great Trigonometrical Survey of India chained such triangles across a subcontinent in the 19th century.
The ambiguous case (SSA)
Picture side b running from corner A to corner C, and side a hanging from C like a pendulum, swinging down to meet the line of side c. The shortest distance from C to that line is the height h = b sin A, and comparing a with h and b tells you how many triangles exist:
| Angle A | Condition | Triangles |
|---|---|---|
| Acute | a < h | None (a is too short) |
| Acute | a = h | One, with B = 90° |
| Acute | h < a < b | Two |
| Acute | a ≥ b | One |
| Right or obtuse | a ≤ b | None |
| Right or obtuse | a > b | One |
The calculator’s SSA default, a = 10 m, b = 12 m and A = 40°, lands in the two-triangle row: h = 12 × sin 40° ≈ 7.71 m, which is less than 10, and 10 is less than 12. Then sin B = 12 × sin 40° ÷ 10 ≈ 0.7713, so B ≈ 50.47° or B ≈ 129.53°.
| Measure | Triangle 1 | Triangle 2 |
|---|---|---|
| Angle B | 50.47° | 129.53° |
| Angle C | 89.53° | 10.47° |
| Side c | 15.56 m | 2.83 m |
| Area | 60.00 m² | 10.91 m² |
Both share side a and angle A, so they have the same common ratio, 15.5572, and the same circumradius, about 7.78 m. Only extra information, such as knowing B is obtuse, tells you which one you have. Raise a to 14 and the ambiguity disappears: B ≈ 33.43°, because the alternative 146.57° plus the 40° at A would already exceed 180°.
A little history
A sine rule for triangles drawn on a sphere was known to astronomers in the Islamic world by the 10th century, and the Persian scholar Nasir al-Din al-Tusi set out the plane version clearly in the 13th century. For SAS or SSS, switch to the law of cosines calculator, or use the general triangle calculator.
Frequently asked questions
What is the law of sines?
In any triangle, each side divided by the sine of its opposite angle gives the same number: a ÷ sin A = b ÷ sin B = c ÷ sin C. That common ratio equals 2R, the diameter of the circle through the three corners.
When can I use the law of sines?
You need one side together with its opposite angle, plus one more side or angle. That covers AAS, ASA (find the third angle first, which gives you the pair) and SSA. For SAS or SSS, start with the law of cosines instead.
Why does SSA sometimes give two triangles?
Solving for angle B means taking an inverse sine, and every sine value between 0 and 1 belongs to two angles, B and 180° − B. When side a is longer than the height b sin A but shorter than b, both angles leave room for a third angle, so two different triangles fit.
How do I know there is no triangle at all?
If angle A is acute and side a is shorter than b sin A, side a cannot reach the third side. If A is 90° or more, side a must be longer than side b. The calculator tells you which condition failed.
Can I enter angles in radians?
Yes. Switch the Angle unit to radians and type the angles in radians. Results are then shown in radians with the degree value underneath.