Law of Sines Calculator
Find a triangle's missing side or angle with the law of sines, including both answers in the ambiguous case.
Change the values and press Calculate to work out your own figures.
Missing side (two angles and a side)
Enter side a, its opposite angle A and angle B.
Show the working
- A side and its opposite angle are known: the rest from the law of sines, a ÷ sin A = b ÷ sin B = c ÷ sin C
- Area = √(s(s − a)(s − b)(s − c)) = 38.30222216
Missing angle (two sides and an angle)
Enter sides a and b and angle A (opposite a).
Show the working
- A side and its opposite angle are known: the rest from the law of sines, a ÷ sin A = b ÷ sin B = c ÷ sin C
- Area = √(s(s − a)(s − b)(s − c)) = 37.22368524
How to use it
Enter side a, its opposite angle A and angle B. Enter sides a and b and angle A (opposite a).
Enter the values you know (decimals, fractions, surds such as 2√5, or angles such as π/3 in radians). Show the working gives the rule used at each step.
Key facts
a ÷ sin A = b ÷ sin B = c ÷ sin C
One, two or no triangles
Given a, b and A with a < b, the triangle exists only if a ≥ b sin A, and then there are two when a < b.
Questions
What is side b with a = 10, A = 65° and B = 50°?
≈ 8.452365235.
Why two answers?
sin B = sin(180° − B), so an obtuse B can fit as well as an acute one.
Formulas
a ÷ sin A = b ÷ sin B = c ÷ sin C
Sources
Limitations
- Angles can be typed in degrees or radians; radians may be written with π (π/3, 2pi/3).