Law of Sines Calculator — Solve Any Triangle (AAS, ASA, SSA & Ambiguous Case)

Law of Sines Formula

The law of sines states that the proportion between the length of a side of a triangle to the sine of the opposite angle is equal for each side:

a / sin(α) = b / sin(β) = c / sin(γ)

This ratio is also equal to the diameter of the triangle's circumscribed circle (circumcircle). Unlike the Pythagorean theorem, the law of sines applies to any triangle — not just right triangles.

Law of Sines Application

You can use the law of sines to solve triangulation problems in two main situations:

• AAS / ASA — Two angles and one side. Given any two angles (α and β) and one known side, you can immediately compute the third angle (γ = 180° − α − β) and then all remaining sides.
• SSA — Two sides and a non-included angle (Ambiguous Case). Given sides a and c with angle α opposite side a, there may be zero, one, or two valid triangles. The ambiguous case arises when:
  • Angle α is acute (α < 90°);
  • Side a is shorter than side c (a < c);
  • Side a is longer than the altitude h = c × sin(α).
  Our calculator detects this automatically and shows both solutions when they exist.

Law of Sines Calculator — How to Use It?

1. Choose a mode: AAS/ASA if you know two angles and one side; SSA if you know two sides and the angle opposite one of them.
2. Select your unit system — Metric (cm, m, km) or US customary (in, ft, yd) — and pick the specific length unit.
3. Enter the known values in the form on the left. Use the triangle diagram (α opposite a, β opposite b, γ opposite c) as a reference.
4. Click Calculate and the calculator will instantly display all sides, angles, circumradius, perimeter, and area.

FAQs

Can I use the law of sines on right triangles?

Yes. The law of sines works for all triangles, including right triangles. You need either two sides and an angle opposite one of them, or two angles and one side.

When should I use the law of sines vs. the law of cosines?

Use the law of sines when you know:

• Two angles and one side (AAS or ASA), or
• Two sides and an angle opposite one of those sides (SSA).

Use the law of cosines when you know:

• Three sides (SSS), or
• Two sides and the included angle (SAS).

How do I find an unknown side using the law of sines?

To find side a given side b and angles α and β:

a = b × sin(α) / sin(β)

What is the circumradius?

The circumradius R is the radius of the circle that passes through all three vertices of the triangle. It follows directly from the law of sines:

R = a / (2 × sin(α))