# Law of Sines Calculator

Solve triangles using the law of sines. Enter one side, its opposite angle, and another angle to find all remaining sides and angles.

## What this calculates

Solve any triangle using the law of sines. Enter a known side, its opposite angle, and one other angle to calculate all remaining sides, angles, and the area of the triangle.

## Inputs

- **Side a** — min 0 — The length of the known side.
- **Angle A (degrees)** — min 0, max 180 — The angle opposite side a, in degrees.
- **Angle B (degrees)** — min 0, max 180 — The second known angle, in degrees.

## Outputs

- **Side b** — The side opposite angle B.
- **Side c** — The side opposite angle C.
- **Angle C** — The third angle (180 - A - B).
- **Area** — The area of the triangle.

## Details

The law of sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This relationship allows you to find unknown sides or angles when you know at least one side-angle pair and one additional piece of information.

This calculator handles the AAS (angle-angle-side) case, where you know two angles and a non-included side. Since the three angles of a triangle sum to 180 degrees, knowing two angles immediately gives you the third. Then the law of sines finds the remaining sides.

The law of sines is particularly useful in surveying (triangulation), navigation (finding distances to landmarks), astronomy (calculating stellar distances), and engineering (structural analysis). When combined with the law of cosines, it can solve any triangle completely.

## Frequently Asked Questions

**Q: What is the law of sines?**

A: The law of sines states that in any triangle, the ratio of each side to the sine of its opposite angle is equal: a/sin(A) = b/sin(B) = c/sin(C). This constant ratio equals the diameter of the circumscribed circle (2R).

**Q: When should I use the law of sines?**

A: Use the law of sines when you know: (1) two angles and any side (AAS or ASA), or (2) two sides and an angle opposite one of them (SSA, the ambiguous case). For two sides and the included angle (SAS) or three sides (SSS), use the law of cosines instead.

**Q: What is the ambiguous case of the law of sines?**

A: The ambiguous case (SSA) occurs when you know two sides and a non-included angle. There may be zero, one, or two valid triangles. This happens because sin(x) = sin(180-x), so the inverse sine can give two possible angles. This calculator avoids the ambiguous case by requiring two angles instead.

**Q: How do I find the area using the law of sines?**

A: Once all sides and angles are known, the area can be calculated as Area = 0.5 x a x b x sin(C), where C is the angle between sides a and b. Alternatively, Area = 0.5 x b x c x sin(A) or 0.5 x a x c x sin(B). All three formulas give the same result.

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Source: https://vastcalc.com/calculators/math/law-of-sines
Category: Math
Last updated: 2026-04-21