Example 1
- a = 10
- α = 30°
- β = 90°
20.000000
What is b with 10, 30° and 90°? 20.000000. Right-angled, 10 is opposite 30°.
Law of sines calculator
Type side a and the angles α and β in degrees. α sits opposite a, β opposite the unknown b. With a = 10, α = 30° and β = 90° you get b = 20.000000.
The calculator finds one side, not an angle. α in the second field is opposite a. When sin(α) = 0, the result is 0, not a blank.
Results
Enter data and click Calculate.
How it works
The law of sines says a side over the sine of the opposite angle is constant: a/sin α = b/sin β. The calculator rearranges that to b = a · sin(β)/sin(α). With a = 10, α = 30° and β = 90° you have 10 · 1 / 0.5 = 20. The result is 20.000000.
Field a is the known side. Field b is angle α opposite that side, in degrees. Field c is angle β opposite the unknown side. The result label says “Side b”.
When α = β the sides match. 10, 30°, 30° gives 10. When α = 30° and β = 45°, with a = 8 you get 11.313708.
sin(0°) = 0, so α = 0 is a divide by zero. The implementation returns 0. That is not a length of zero, only the branch at a zero sine.
The law of cosines finds a side from two sides and the included angle. The third angle from 180° − α − β is on the neighbouring page. Here you always stay with a sine ratio.
10, 30°, 90° gives 20.000000. 10, 30°, 30° gives 10. 8, 30°, 45° gives 11.313708.
Formula
b = a · sin(β°) / sin(α°), when sin(α) ≠ 0; otherwise 0
How to use
- Type side a, for example 10.
- Type α opposite a, for example 30, and β opposite the unknown b, for example 90.
- Click Calculate. b comes out 20.000000.
- When α = 0, the result is 0, because sin(0°) = 0.
- A side from two sides and the included angle lives on the law of cosines.
10, 30°, 90° gives 20.000000
b = a · sin(β)/sin(α). With a = 10, α = 30° and β = 90° side b is 20.000000. α = 0 returns 0.
- sines
- The side-to-sine ratio. 10 and 30° with 90° give 20.000000.
- α
- The angle opposite the known side a. Second field. 0° returns 0, because sin 0° = 0.
- β
- The angle opposite the unknown b. Third field. 90° with a = 10 and α = 30° gives 20.000000.
Examples
Example 2
- a = 8
- α = 30°
- β = 45°
11.313708
What is b with 8, 30° and 45°? 11.313708.
Example 3
- a = 10
- α = 30°
- β = 30°
10
Do equal angles give equal sides? Yes. 10 and 10.
Related calculators
Common questions
What is b with a = 10, α = 30° and β = 90°?
20.000000. 10 / sin 30° = b / sin 90°, so b = 20.
What does the law of sines mean?
A side over the sine of the opposite angle is the same number. a/sin α = b/sin β.
Which field is the angle opposite a?
The second, α. The third is β opposite the unknown b.
What if α = 0°?
sin(0°) = 0. The calculator returns 0, not a blank.
Does 10, 30°, 30° give 10?
Yes. Equal angles, equal sides.
What about 8, 30° and 45°?
11.313708. 8 · sin 45° / sin 30°.
Does this find an angle?
No. It finds side b. The angle from 180° lives on the sum-of-angles card.
How is this different from the law of cosines?
Here a sine ratio and one side. There, two sides and the included angle.
Do the angles have to be in degrees?
Yes. The calculator multiplies by π/180 before the sine.
Does the order of α and β matter?
Yes. Swap them and you get another side, because another angle sits opposite a.
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.