Quick answer
Solve a triangle from two angles and one side (law of sines). Enter angle A, side a and angle B and the law of sines calculator returns side b, angle C and side c.
Related tools: Sine Rule Calculator: Apply the sine rule to find a missing side or angle of a triangle from an angle and its opposite side. Law of Cosines Calculator: Third side of a triangle from two sides and the included angle (law of cosines). Trigonometry Calculator: All six trigonometric functions of an angle.
This page covers the same calculation as the sine rule calculator page; the two give identical results.
How to use the Law of Sines Calculator
- Enter angle A (°) (between 0 and 179).
- Enter side a (opposite A) (above 0).
- Enter angle B (°) (between 0 and 179).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Angle A | A number, in ° | > 0, ≤ 179 |
| Side a (opposite A) | A number | > 0 |
| Angle B | A number, in ° | > 0, ≤ 179 |
Formula and method
Formula
a ÷ sin A = b ÷ sin B = c ÷ sin C
IEEE-754 double precision; shown to at most 6 decimal places.
Third angle C = 180° − A − B, then each side from the common ratio. Requires A + B < 180°.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Angle A = 30; Side a (opposite A) = 5; Angle B = 90.
| Result | Value |
|---|---|
| Side b | 10 |
| Angle C | 60° |
| Side c | 8.660254 |
The first line is the headline result (side b). Below it the law of Sines Calculator lists angle C and side c, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
How the result changes with angle A
Holding every other input at the first worked example, this table changes only angle A. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Angle A | Side b | Angle C | Side c |
|---|---|---|---|
| 15 | 19.318517 | 75° | 18.660254 |
| 30 | 10 | 60° | 8.660254 |
| 60 | 5.773503 | 30° | 2.886751 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Angle B: entering "80" shows "Angles A and B must add up to less than 180°." beside that field and no result.
Choosing between the Law of Sines Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Sine Rule Calculator](/math/sine-rule-calculator/) | angle A, side a and angle B | Side b |
| [Law of Cosines Calculator](/math/law-of-cosines-calculator/) | side b, side c and included angle A | Side a |
| [Trigonometry Calculator](/math/trigonometry-calculator/) | angle and unit | sin |
| [SOHCAHTOA Calculator](/math/sohcahtoa-calculator/) | acute angle, known side length and known side is the | Opposite |
| [Reference Angle Calculator](/math/reference-angle/) | angle | Reference angle |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- A + B must be less than 180°
- The ambiguous SSA case is not handled
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for angle B to confirm this.
Frequently asked questions
What does the Law of Sines Calculator calculate?
Solve a triangle from two angles and one side (law of sines). It returns Side b, Angle C and Side c.
What formula does the Law of Sines Calculator use?
a ÷ sin A = b ÷ sin B = c ÷ sin C. Third angle C = 180° − A − B, then each side from the common ratio.
What do I need to enter in the law of Sines Calculator?
You enter Angle A (°), Side a (opposite A) and Angle B (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. A + B must be less than 180°.
What are the limits of this calculator?
A + B must be less than 180°. The ambiguous SSA case is not handled.
What is an example of a law of Sines result?
With angle A 30, side a (opposite A) 5 and angle B 90, the side b is 10.
Where does the method come from?
The method follows: Standard mathematical definition.
Related calculators
- Sine Rule Calculator — Apply the sine rule to find a missing side or angle of a triangle from an angle and its opposite side.
- Law of Cosines Calculator — Third side of a triangle from two sides and the included angle (law of cosines).
- Trigonometry Calculator — All six trigonometric functions of an angle.
- SOHCAHTOA Calculator — Find the other two sides of a right triangle from one side and an acute angle.
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
- Inverse Sine Calculator — arcsin (inverse trigonometric function) with the result in degrees and radians.
Sources
- Standard mathematical definition
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.