Law of Sines Calculator
Solve a triangle from two angles and a side using the law of sines — remaining sides, the third angle and the area.
How it works
a ÷ sin A = b ÷ sin B = c ÷ sin C · C = 180° − A − B · Area = ½ · a · b · sin C
The law of sines says each side of a triangle divided by the sine of its opposite angle gives the same constant. Knowing two angles and one side, you first get the third angle (angles sum to 180°), then scale the known side by the ratio of sines to find the others. It’s the go-to tool for the angle-angle-side and angle-side-angle cases.
Worked example
With A = 40°, a = 10 and B = 60°, the third angle is C = 180 − 40 − 60 = 80°. Then b = 10 × sin 60° ÷ sin 40° ≈ 13.4680 and c = 10 × sin 80° ÷ sin 40° ≈ 15.3209. The area is ½ × 10 × 13.4680 × sin 80° ≈ 66.3216 square units.
Sources: Law of sines — Wikipedia
Frequently asked questions
When should I use the law of sines versus the law of cosines?
Use the law of sines when you know an angle and its opposite side (plus one more angle or side). Use the law of cosines when you know two sides and the included angle, or all three sides — cases where no angle-side pair is complete.
What is the ambiguous case?
If you know two sides and a non-included angle (SSA), there can be zero, one or two valid triangles. This calculator takes two angles and a side, which always gives a single triangle, so it avoids that ambiguity.
Why must the two angles total less than 180°?
A triangle’s three angles sum to exactly 180°. If A and B already reach or exceed 180°, no positive angle C is left, so the shape can’t close into a triangle.
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This calculator is for educational purposes. Double-check important results before acting on them.