Law of Sines Calculator

Use the law of sines to find a missing side from two angles and a side, or a missing angle from two sides and a non-included angle.

Side b (opposite B) —

How to Use the Law of Sines

Choose what you know: two angles and a side, or two sides and an angle.

1

Two angles + a side (AAS)

Enter two angles and the side opposite the first to find the side opposite the second with a ⁄ sin(A) = b ⁄ sin(B).

2

Two sides + an angle (SSA)

Enter two sides and the angle opposite the first to find the angle opposite the second.

3

Read the result

The unknown side or angle appears immediately; SSA can have a second (obtuse) solution, which the steps note.

Frequently Asked Questions

What is the law of sines?
The law of sines states that in any triangle the ratio of each side to the sine of its opposite angle is the same: a ⁄ sin(A) = b ⁄ sin(B) = c ⁄ sin(C). That shared ratio equals the diameter of the triangle’s circumscribed circle. Knowing any three of those four quantities lets you cross multiply to find the fourth. For A = 30°, a = 5 and B = 45°, b = 5 × sin(45°) ⁄ sin(30°), which is about 7.071.
When do you use the law of sines?
Use the law of sines when you know two angles and any side (AAS or ASA), or two sides and an angle opposite one of them (SSA). In the AAS and ASA cases the third angle comes free from the 180° sum, so the triangle is fully determined. For two sides and the included angle (SAS), or all three sides (SSS), use the law of cosines instead. The SSA case is the one that can be ambiguous.
What is the ambiguous case?
In the SSA case, two sides and a non-included angle, there can be zero, one or two triangles that fit the measurements. The reason is that sin(θ) = sin(180° − θ), so both an acute angle and its obtuse supplement can satisfy the ratio. This calculator returns the acute angle; check whether 180° minus that angle still leaves a positive third angle, because if it does a second valid triangle exists. No triangle exists at all when the computed sine comes out above 1.
Is my data sent anywhere?
Nothing you type is sent anywhere. The law of sines calculator runs entirely in your browser with JavaScript, so your sides and angles never leave your device. There is no account, no upload step and nothing is logged. The page keeps working offline once it has loaded.

Related geometry calculators

Circle Area Calculator

Area of a circle from radius or diameter.

Geometry

→

Sphere Area Calculator

Surface area and volume of a sphere.

Geometry

→

Polygon Area Calculator

Area of a polygon from its coordinates.

Geometry

→

Volume Calculator

Volume of a cube, box, sphere, cylinder, cone or pyramid.

Geometry

→

Surface Area Calculator

Surface area of common 3-D solids.

Geometry

→

Distance Formula Calculator

Distance between two points.

Geometry

→

Area of a Triangle Calculator

Triangle area from base, sides or coordinates.

Geometry

→

Law of Cosines Calculator

Solve a triangle with the law of cosines (SAS / SSS).

Geometry

→

All Math Calculators

Algebra, matrices, statistics and geometry with step-by-step solutions.

Math

→