Law of Sines Calculator
Enter two angles and one side, and find the remaining sides and angle using the law of sines.
How to use the law of sines calculator
- Enter angle A and angle B, in degrees.
- Enter side a, which is opposite angle A.
- Angle C, side b and side c update automatically.
What is the law of sines?
The law of sines describes the relationship between the sides and angles in any triangle: a/sin(A) = b/sin(B) = c/sin(C). It's used to solve triangles where you know two angles and one side (AAS/ASA), or two sides and an angle not between them (SSA).
The ambiguous case
If you know two sides and an angle that does not lie between them, there may be two different triangles matching the same measurements. This is known as the ambiguous case, and it arises because the sine of an angle and the sine of its supplement are identical — sin(40°) and sin(140°) give exactly the same value. The law of sines alone therefore cannot distinguish the two solutions.
In practice that means an answer should be checked against the geometry. If the given angle is large enough, only one triangle is possible; if it is small and the sides are of suitable length, two exist, and the context has to decide which is correct. If you know two angles and a side, or two sides and the angle between them, the solution is always unique.
Worked example: with angle A = 30°, angle B = 60° and side a = 5, first find the third angle: C = 180° − 30° − 60° = 90°. The law of sines then gives b = 5 × sin(60°) ÷ sin(30°) = 8.66 and c = 5 × sin(90°) ÷ sin(30°) = 10. The triangle is therefore right-angled, and the sides 5, 8.66 and 10 are exactly the ratio 1 : √3 : 2 that characterises a 30-60-90 triangle.
Frequently asked questions
When do I use the law of sines instead of the law of cosines?
Use the law of sines when you know at least one angle-side pair (an angle and the side opposite it), such as two angles and one side. Use our law of cosines calculator when you instead know two sides and the angle between them, or all three sides.
Does the law of sines work for all triangles?
Yes, the law of sines applies to all triangles — not just right triangles, unlike basic trigonometry. That's exactly why it's so useful for solving arbitrary triangles.
What is the law of sines used for in practice?
Among other things, in navigation, surveying and astronomy — anywhere you need to calculate distances or angles in triangles that aren't right triangles, such as with triangulation.