# Law of Sines Calculator

URL: https://speedycalculator.com/calculators/math/law-of-sines/
Last updated: 2026-10-25

> The law of sines calculator finds an unknown side of a triangle from one side and two angles. It also gives the third angle, the remaining side and the circumradius.

## Inputs

- Side a
- Angle A (opposite a) (°)
- Angle B (°)

## Results

- Angle C
- Side b
- Side c
- Circumradius

## Formula

```
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R
```

## Explanation

In any triangle, each side is proportional to the sine of its opposite angle.

### How to use the Law of Sines Calculator

- **Side a**
- **Angle A (opposite a)**
- **Angle B**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Find the third angle from 180°, then scale the known side by the ratio of sines.

### Inputs explained

FieldWhat to enterSide aA numberAngle A (opposite a)In °Angle BIn °

### What each result means

- **Angle C** = 180 - [Angle A (opposite a)] - [Angle B]
- **Side b** = [Side a] × sin(rad([Angle B])) ÷ sin(rad([Angle A (opposite a)]))
- **Side c** = [Side a] × sin(rad([Angle C])) ÷ sin(rad([Angle A (opposite a)]))
- **Circumradius** = [Side a] ÷ (2×sin(rad([Angle A (opposite a)])))

### Example values

How the main result changes when **Side a** changes and everything else stays at the example values above:

Side aSide b57.51367.511.27041015.027112.518.78391522.54072030.0543

## Worked example

Inputs — Side a: 10; Angle A (opposite a): 40 °; Angle B: 75 °.

Results — Angle C = 65 °; **Side b = 15.027138**; Side c = 14.099646; Circumradius = 7.778619.

## FAQ

**What is the ambiguous case?**
With two sides and a non-included angle, two different triangles may fit.

**Do the angles need to add to 180°?**
Yes; if A + B ≥ 180°, no triangle exists.

**What formula does the Law of Sines Calculator use?**
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R.