What is the law of sines?
The law of sines connects each side of a triangle with the sine of its opposite angle. For every valid triangle, the three side-to-sine ratios are equal.
Here, a, b, and c are opposite angles A, B, and C. The common value is the diameter of the triangle's circumcircle, where R is the circumradius.
The law works for acute, right, and obtuse triangles. A right triangle is not a special exception; it is simply one of the valid shapes for which the same ratio remains true.
Law of sines formulas
Solving for a side
Solving for an angle
CalculatorSoup lists the cyclic versions of these equations for all three sides and all three angles. The exact form you use depends on which known side-angle pair is opposite each other.
Why does the sine rule work?
Draw the circumcircle through the three vertices of a triangle. Each side is a chord of that circle. A chord's length is related to the sine of the central angle it subtends, which leads to the common relationship a = 2R sin A, and similarly for b and c.
This interpretation explains why the ratio is constant: all three sides are chords of the same circumcircle. Omni Calculator likewise describes the common ratio as the diameter of the circumcircle.
AAS, ASA, and SSA: when the law of sines applies
AAS — angle, angle, side
Know two angles and a side opposite one of them. Find the third angle from A + B + C = 180°, then use the sine rule for the remaining sides.
ASA — angle, side, angle
Know two angles and the side between them. Find the third angle first, then use the sine rule to determine the other sides.
SSA — side, side, angle
Know two sides and an angle opposite one of them. The inverse sine can create an ambiguous case, so the data may describe zero, one, or two triangles.
When cosine is better
For SSS or SAS data, the law of cosines is usually the direct relationship: three sides, or two sides with their included angle.
The AAS and SSA distinctions, including the possible two-solution SSA case, are emphasized by Omni Calculator and CalculatorSoup.
Step-by-step examples
Example 1 — AAS: find two missing sides
Suppose a = 7, A = 40°, and B = 60°.
b = 7 · sin60° / sin40° ≈ 9.436
c = 7 · sin80° / sin40° ≈ 10.716
The triangle is now completely determined: all three angles and all three sides are known.
Example 2 — SSA: an ambiguous case
Take a = 8, b = 10, and A = 40°. First calculate:
sin B = 10 · sin40° / 8 ≈ 0.803
B₁ ≈ 53.4°
Because sine has the same value at an angle and its supplement, a second candidate is B₂ = 180° − B₁ ≈ 126.6°. Since A + B₂ < 180°, both configurations are geometrically possible. This is the classic ambiguous case.
Why SSA can produce two triangles
In an SSA problem, you know an angle and its opposite side plus another side. If the known angle is acute, the second side can sometimes intersect the ray in two different positions. Algebraically, this happens because sin θ = sin(180° − θ).
Compute sine
Use sin B = b sin A / a.
Find B₁
Take arcsin of the valid ratio.
Test B₂
Try 180° − B₁.
Check the sum
Keep a solution only when A + B < 180°.
Right triangles and the 30°–60°–90° triangle
Right triangle
The law of sines still works. If C = 90°, then c is the hypotenuse and c / sin90° = c = 2R.
30°–60°–90°
If a is opposite 30°, b opposite 60°, and c opposite 90°, then a / ½ = b / (√3/2) = c. Therefore c = 2a and b = √3a.
Omni Calculator uses the 30°–60°–90° triangle to illustrate how the sine rule turns known side lengths into exact relationships.
How to check a sine-rule solution
A valid triangle must satisfy the angle sum and positive side conditions:
For an inverse-sine calculation, the ratio passed to arcsin must lie in the interval −1 ≤ x ≤ 1. For a triangle, the resulting angles must also leave a positive third angle.
| Check | Meaning | Expected |
|---|---|---|
| A+B+C | Triangle angle sum | 180° |
| sin target | Inverse-sine input | −1 to 1 |
| a,b,c | Side lengths | Positive |
| a / sin A | Common sine ratio | = b/sin B = c/sin C |