Case 1
- Side a
- 10
- Angle A, opposite side a
- 30°
- Angle B, opposite side b
- 45°
14.1421
Open with these valuesLaw of Sines Calculator
14.1421units
Result: 14.1421 unitsEvery side of a triangle stands in the same ratio to the sine of its opposite angle, so b = a × sin B ÷ sin A. A side of 10 opposite 30°, with 45° opposite the side you want, gives 10√2 — about 14.142. The two angles have to add to less than 180°.
The numbers at a glance
Held fixed: Side a 10.000, Angle A, opposite side a 30.00 °.
| Angle B, opposite side b (°) | Result |
|---|---|
| 20.00 | 6.8404 |
| 40.00 | 12.8558 |
| 45.00Your value | 14.1421 |
| 60.00 | 17.3205 |
| 80.00 | 19.6962 |
Worked examples
Case 2
- Side a
- 7
- Angle A, opposite side a
- 40°
- Angle B, opposite side b
- 60°
9.4311
Open with these valuesCase 3
- Side a
- 8
- Angle A, opposite side a
- 60°
- Angle B, opposite side b
- 90°
9.2376
Open with these valuesHow it's calculated
b = a × sin B ÷ sin A
- StepEnter the known side a and the angle A that lies opposite it.
- StepEnter angle B, the one opposite the side you are looking for.
- ResultRead side b — A and B together must stay below 180°.
What this number means
Degrees go in, not radians
Type 30 for thirty degrees; the conversion to radians sits inside the formula. Each angle has to stay above 0° and below 180°.
A and B together must stay under 180°
The third angle is whatever is left of 180°, so at A + B ≥ 180° there is no triangle at all. The formula still hands back a ratio, but that number is not the side of anything.
The ambiguous SSA case cannot arise here
Two sides and an angle that is not between them can fit two different triangles, because the arcsine only ever returns the acute angle and quietly drops the obtuse twin. This calculator starts from one side and two angles instead, and that fixes exactly one triangle — so it never has to choose.
Equal angles, equal sides
With A = B the two sines cancel and side b comes back equal to side a: 12 in, 12 out at 35° and 35°. That is the isosceles triangle falling out of the ratio in one line.
Commonly misread
Angle A can be any angle of the triangle.
A has to be the angle opposite the side a you entered, and B the one opposite the side you are after. Pair a side with the wrong angle and the result belongs to no triangle.
Enter 0.5236 for 30°, because the sine needs radians.
Enter 30. The radian conversion already happens inside the formula, so 0.5236 would be read as roughly half a degree.
A = 120° with B = 70° is fine — a number comes back.
Those two add to 190°, leaving nothing for the third angle, so no such triangle exists. What comes back is a bare ratio, not a side length.
Reference table
| Side a and angles A, B | Exact | Side b |
|---|---|---|
| 5, 90, 30 | 5 ÷ 2 | 2.5000 |
| 7, 40, 60 | 7·sin 60° ÷ sin 40° | 9.4311 |
| 8, 60, 90 | 16 ÷ √3 | 9.2376 |
| 10, 30, 45 | 10√2 | 14.1421 |
| 10, 30, 60 | 10√3 | 17.3205 |
| 12, 35, 35 | equal angles, equal sides | 12.0000 |
Questions
How do I find a side with the law of sines?
Use the ratio b ÷ sin B = a ÷ sin A, which rearranges to b = a × sin B ÷ sin A. Enter the known side a, the angle A opposite it, and the angle B opposite the side you want. A side of 10 with A = 30° and B = 45° gives 10 × sin 45° ÷ sin 30° ≈ 14.142.
Should I enter angles in degrees or radians?
In degrees — the conversion to radians happens inside the formula. Each angle must be greater than 0° and less than 180°.
What if the two angles add to 180° or more?
Then no triangle exists, because nothing is left over for the third angle. The formula still returns a ratio, but that number is not the side of any triangle. Keep A + B below 180°.
When do I use the law of sines instead of the law of cosines?
Use the law of sines when you know an angle and the side opposite it, plus one more angle or side. Use the law of cosines when you know two sides and the angle between them, or all three sides — cases where no side sits opposite a known angle.
What is the ambiguous SSA case?
When you know two sides and an angle that is not between them, the law of sines can allow zero, one or two triangles. This calculator avoids that ambiguity by working from one side and two angles, which always fixes a single triangle.
Sources and last check
Information, not professional advice.