Quick answer
The sine rule states a ÷ sin A = b ÷ sin B = c ÷ sin C. With A = 30°, a = 5 and B = 90°, b = 10, C = 60° and c = 8.6603.
Enter two angles and the side opposite one of them. The calculator returns the remaining sides and angle.
Related tools: 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. Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides.
How to use the Sine Rule 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 sine Rule 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.
Worked example
a ÷ sin A = 5 ÷ 0.5 = 10, so b = 10 × sin 90° = 10 and c = 10 × sin 60° = 8.6603. C = 180° − 30° − 90° = 60°.
Where this estimate stops applying
A + B must be less than 180°. The ambiguous SSA case, with two sides and a non-included angle, can have two solutions and is not handled. Use the law of cosines for SAS or SSS.
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 Sine Rule 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 |
|---|---|---|
| [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 |
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Law of Sines Calculator](/math/law-of-sines-calculator/) | angle A, side a and angle B | Side b |
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
When do I use the sine rule?
Use the sine rule when you know two angles and a side (AAS or ASA).
What is the SSA ambiguity?
Two triangles may fit.
What is the 'k' in the rule?
The common ratio, here 10, equal to the circumdiameter.
What does the Sine Rule Calculator calculate?
Apply the sine rule to find a missing side or angle of a triangle from an angle and its opposite side. It returns Side b, Angle C and Side c.
What formula does the Sine Rule 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 sine Rule 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 sine Rule 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: OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions; OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry.
Related calculators
- 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.
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Law of Sines Calculator — Solve a triangle from two angles and one side (law of sines).
- Sine Calculator — sin of an angle in degrees or radians.
- Inverse Sine Calculator — arcsin (inverse trigonometric function) with the result in degrees and radians.
- Cosine Rule Calculator — Apply the cosine rule to find a missing side or angle of a triangle from the sides and the included angle.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.