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

Solve Any Triangle

Select your known values (SSS, SAS, ASA, AAS, or SSA) and enter measurements to solve the triangle and compute area, perimeter, inradius, and more.

Documentation

This calculator is also known as Law of Sines Calculator.

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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.