Math

Law of Sines Calculator

Solving Triangles with the Law of Sines

The Law of Sines states that a/sin A = b/sin B = c/sin C for any triangle. Use ASA or AAS mode to find missing sides, or SSA mode to explore the ambiguous case where two triangles may be valid.

Triangle Solver Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Right Triangle

Classic 3-4-5 right triangle using SSS mode.

Key values: a = 3 · b = 4 · c = 5

Roof Pitch

Calculate a roof triangle with two sides and a 35-degree pitch angle.

Key values: a = 6 · b = 8 · C = 35°

Equilateral Triangle

A perfect equilateral triangle with all sides equal to 10.

Key values: a = 10 · b = 10 · c = 10

Documentation

This calculator is also known as Law of Sines Calculator.

Read the complete guide

The Law of Sines Formula

For any triangle with sides a, b, c opposite to angles A, B, C: a/sin A = b/sin B = c/sin C. This ratio equals the diameter of the circumscribed circle (2R).

When to Use Law of Sines vs. Law of Cosines

Use Law of Sines when you have:

ASA: two angles and the included side
AAS: two angles and any side
SSA: two sides and a non-included angle (ambiguous case)

Examples

Navigation Bearing (AAS)

A ship travels on bearing 040°. After 10 km, it turns to bearing 110°. The angle at the starting point is 70° and at the turning point is 70°. Find the direct return distance.

C = 180 - 70 - 70 = 40°. c = 10 × sin(40°)/sin(70°) ≈ 6.84 km.

Key takeaway: Law of Sines is ideal for navigation problems where angles are measured from bearings.

Surveying: ASA

A surveyor measures a plot. Angle A = 55°, angle B = 75°, side c (between A and B) = 120 m. Find the other two sides.

C = 50°. a = 120 × sin(55°)/sin(50°) ≈ 128.3 m. b = 120 × sin(75°)/sin(50°) ≈ 151.2 m.

Key takeaway: ASA is the most common surveying scenario when you can measure two angles from a known baseline.

SSA Ambiguous Case

Given a = 7, b = 10, A = 30°. How many triangles exist?

sin B = 10 × sin(30°)/7 = 0.714. B₁ = 45.6°, B₂ = 134.4°. Both give valid C > 0, so two triangles exist.

Key takeaway: The SSA ambiguous case always requires checking whether sin B ≤ 1 and whether the supplement also forms a valid triangle.

Applying the Law of Sines

Follow these steps for reliable solutions:

Identify your known information: ASA, AAS, or SSA
For SSA, always check sin B ≤ 1 first to confirm a triangle exists
Check the supplementary angle (180° - B) to detect the ambiguous case
Verify your answer: angles must sum to 180° and satisfy the triangle inequality

Frequently Asked Questions about Law of Sines Calculator

What is the Law of Sines?

The Law of Sines states that the ratio of a side to the sine of its opposite angle is constant for all three sides: a/sin A = b/sin B = c/sin C. This common ratio equals the diameter of the triangle's circumscribed circle (2R).

When does the SSA ambiguous case give two solutions?

When the angle A is acute and the side opposite it (a) is shorter than the adjacent side (b), sin B may be less than 1, allowing both B and 180° - B as valid angles. Check whether 180° - B - A is still positive to confirm two solutions.

Can the Law of Sines solve all triangles?

No. The Law of Sines requires at least one angle-side pair. For SSS (three sides, no angles) or SAS (two sides and the included angle), use the Law of Cosines instead.