What is the Law of Sines?

The Law of Sines states that for any triangle with sides a, b, c opposite to angles A, B, C respectively: a/sin(A) = b/sin(B) = c/sin(C). This common ratio equals the diameter of the triangle's circumscribed circle (circumradius × 2). The law holds for all triangles — acute, right, and obtuse — not just right triangles.

When do you use the Law of Sines?

Use the Law of Sines when you know: (1) two angles and any side (AAS or ASA cases), or (2) two sides and a non-included angle (SSA case, which is ambiguous). You cannot use the Law of Sines when you know only three sides (SSS) or two sides and the included angle (SAS) — those require the Law of Cosines. A simple rule: if you have at least one angle-side opposite pair, the Law of Sines applies.

What is the ambiguous case (SSA)?

The SSA (side-side-angle) case is ambiguous because knowing two sides and a non-included angle may produce 0, 1, or 2 valid triangles. For example, if A = 35°, a = 6, b = 8: sin(B) = 8·sin(35°)/6 ≈ 0.765, giving B ≈ 49.9° or B ≈ 130.1°. Both yield valid triangles with different shapes. If sin(B) > 1, no triangle exists; if sin(B) = 1, exactly one (right triangle); otherwise, check both B and 180°−B.

How do you find the area using the Law of Sines?

Once you know all three sides and angles, use Area = ½ × a × b × sin(C), where a and b are any two sides and C is the included angle between them. This formula works for any triangle. Alternatively, Area = (a² × sin(B) × sin(C)) / (2 × sin(A)) using only angles and one side, derived from the Law of Sines. Both formulas are equivalent when the triangle is fully solved.

What is the difference between Law of Sines and Law of Cosines?

The Law of Sines uses ratios of sides to sines of opposite angles and is best for AAS, ASA, and SSA problems. The Law of Cosines (c² = a² + b² − 2ab·cos(C)) relates all three sides and one angle, and is best for SAS (two sides plus included angle) and SSS (all three sides) problems. The Law of Cosines reduces to the Pythagorean theorem when C = 90°. Use sines for angle-rich problems; use cosines for side-rich problems.

How do you solve an AAS triangle?

With two angles (A and B) and one non-included side (a opposite angle A): first find the third angle C = 180° − A − B. Then use the sine ratio: b = a × sin(B)/sin(A) and c = a × sin(C)/sin(A). The area is ½ × b × c × sin(A). Always verify that A + B + C = 180° exactly. If any angle is negative or zero, the inputs are invalid.

How is the Law of Sines used in navigation and surveying?

Surveyors use the Law of Sines to find distances across inaccessible terrain. By measuring a known baseline and two angles from each end to a distant point, they can calculate the distance to that point via triangulation. This technique was used to map the entire Indian subcontinent in the 19th-century Great Trigonometrical Survey. Modern GPS still uses triangle-based calculations at its core.

What is the circumradius of a triangle and how does it relate to the Law of Sines?

The circumradius R is the radius of the circle passing through all three vertices of a triangle (the circumscribed circle). The Law of Sines reveals that a / sin(A) = b / sin(B) = c / sin(C) = 2R. This means the common sine ratio equals twice the circumradius. For a triangle with side a = 10 and opposite angle A = 30°, the circumradius is R = a / (2 sin(A)) = 10 / (2 × 0.5) = 10.

Law of Sines Calculator — Solve Any Triangle

Solves any triangle using AAS, ASA, or SSA — finds all sides, angles, area, and perimeter, and flags the ambiguous SSA case.

How to Use the Law of Sines Calculator

This law of sines calculator supports AAS, ASA, and SSA configurations — select your known configuration using the mode buttons. Then enter the known angles (in degrees) and sides. The calculator fills in all remaining elements: the third angle, any missing sides, the area (½ab·sin(C)), and the perimeter. For SSA, it detects and reports both solutions if the ambiguous case applies.

For problems where you know two sides and the included angle (SAS), or all three sides (SSS), use our Law of Cosines calculator instead.

The Law of Sines Formula

For any triangle with sides a, b, c and opposite angles A, B, C:

a / sin(A) = b / sin(B) = c / sin(C) = 2R

Where R is the circumradius (radius of the circle passing through all three vertices). This means each side divided by the sine of its opposite angle gives the same constant — a powerful relationship for solving triangles when angle-side pairs are known.

AAS — Two Angles and a Non-Included Side

Given A, B, and side a (opposite to A): find C = 180° − A − B, then b = a·sin(B)/sin(A) and c = a·sin(C)/sin(A). AAS is straightforward — exactly one triangle is possible.

ASA — Two Angles and the Included Side

Given A, C, and side b (between A and C): find B = 180° − A − C, then use the sine ratio. ASA also produces exactly one triangle.

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The Ambiguous SSA Case

SSA (two sides and a non-included angle) is the only case that can produce 0, 1, or 2 valid triangles. Given angle A, opposite side a, and adjacent side b:

Compute sin(B) = b·sin(A)/a
If sin(B) > 1 → no triangle exists (the side a is too short to reach the opposite vertex)
If sin(B) = 1 → exactly one right triangle (B = 90°)
If sin(B) < 1 → two possible angles: B = arcsin(sin(B)) or B = 180° − arcsin(sin(B)). Check whether each leads to a valid triangle (all angles positive and summing to 180°)

The second solution only exists if A is acute and a < b. This ambiguity is a fundamental property of trigonometry — it cannot be resolved without additional information about the triangle.

Finding Triangle Area with the Law of Sines

Once all elements are known, area = ½ × a × b × sin(C) where C is the angle included between sides a and b. If you only have angles and one side, this calculator derives all sides first and then computes the area. For a triangle with A = 40°, B = 60°, a = 10: C = 80°, b = 10·sin(60°)/sin(40°) ≈ 13.47, c = 10·sin(80°)/sin(40°) ≈ 15.32, area = ½ × 10 × 13.47 × sin(80°) ≈ 66.36 square units.

For right triangles specifically, see our Pythagorean theorem calculator which handles those more directly.

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Common Mistakes When Using the Law of Sines

Using Law of Sines for SAS or SSS — you need a known angle-side opposite pair; SAS and SSS require the Law of Cosines
Ignoring the SSA ambiguity — always check whether a second solution exists when given SSA
Mixing up which side is opposite which angle — side a is always opposite angle A, b opposite B, c opposite C
Angle sum error — always verify A + B + C = 180°; rounding errors can accumulate
Using degrees vs radians — make sure your calculator (or this one) is set to degrees when working with angles in degrees

Sources & References

Law of sines — Wikipedia
Solution of triangles — Wikipedia

Law of Sines Calculator