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 constant: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. This lets you solve for unknown sides or angles whenever you have enough information.

When can I use the law of sines?

You need at least one angle-side opposite pair plus one additional piece of information. The two classic cases are AAS/ASA (two angles and any side) and SSA (two sides and the angle opposite the shorter one). For a triangle defined by two sides and the included angle (SAS) or three sides (SSS), use the law of cosines instead.

What is the ambiguous case (SSA)?

When you provide two sides and an angle that is NOT between them, there can be zero, one, or two valid triangles. The calculator checks sin(B) = b*sin(A)/a. If this exceeds 1 there is no solution. If it equals 1 there is exactly one right triangle. Otherwise both B and 180-B may produce a valid triangle, and the calculator shows both.

How is the triangle area calculated?

Once all three sides and angles are known, area = (1/2) * a * b * sin(C). Equivalently you can use any two sides and their included angle. The formula follows directly from the cross-product of two edge vectors.

Does this calculator work for right triangles?

Yes. The law of sines applies to all triangles including right triangles. For a right triangle with C = 90 degrees, sin(C) = 1, so the ratio k = a/sin(A) = c, meaning the hypotenuse equals the diameter of the circumscribed circle.

How accurate are the results?

All calculations use JavaScript 64-bit floating-point arithmetic (IEEE 754 double precision), giving about 15 significant digits. Results are displayed to 6 significant figures, which is more than enough for any practical geometry, surveying, or navigation task.

Law of Sines Calculator

The law of sines is one of the two fundamental identities for solving oblique (non-right) triangles. Given any triangle with sides a, b, c opposite to angles A, B, C respectively, the ratio of each side to the sine of the angle across from it is the same:

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

where R is the circumradius of the triangle. This tool uses that identity to fill in every unknown side, angle, and the area — instantly, in your browser, with no server calls.

How it works

The calculator supports two input modes.

AAS / ASA — two angles plus one side. Because the three interior angles must sum to 180°, knowing two angles immediately gives the third: C = 180 − A − B. Once all angles are known, the common ratio k = (known side) / sin(known angle) is fixed, and every remaining side follows as k × sin(opposite angle).

SSA — two sides and the angle opposite one of them. Here sin(B) = b × sin(A) / a. Three outcomes are possible:

If b × sin(A) / a is greater than 1 — no triangle exists (the side is too short to reach).
If b × sin(A) / a equals 1 — exactly one right triangle (side a is the altitude).
Otherwise — two triangles share the same SSA data. One has the acute angle B, the other the supplementary angle 180° − B. The calculator detects this ambiguous case automatically and shows both solutions.

Worked example

Suppose you measure two angles A = 40° and B = 60° in a surveying triangle, and the side between those measured stations (side c, opposite C) is 8 m.

C = 180 − 40 − 60 = 80°
k = c / sin C = 8 / sin 80° ≈ 8.123
a = k × sin 40° ≈ 8.123 × 0.6428 ≈ 5.22 m
b = k × sin 60° ≈ 8.123 × 0.8660 ≈ 7.03 m
Area = (1/2) × a × b × sin C ≈ (1/2) × 5.22 × 7.03 × sin 80° ≈ 18.1 m²

All of these appear in the result panel alongside the SVG diagram with labelled vertices.

Formula note

The identity a/sin(A) = 2R has an elegant geometric proof: draw the circumscribed circle of the triangle, then use the inscribed-angle theorem which says any inscribed angle is half the central angle subtending the same arc. The sine of the inscribed angle equals the half-chord divided by the radius, giving the relation directly.

For computation the key steps are:

k = knownSide / sin(knownAngle) — the constant ratio
unknownSide = k × sin(unknownAngle)
area = (1/2) × a × b × sin(C) — once all three sides are found

All trigonometric functions operate in degrees; internally the calculator converts to radians for Math.sin and Math.asin, then converts back to degrees for display.