Law of sines calculator

What does the law of sines say?

In every triangle the ratio of a side to the sine of the opposite angle is the same for all three sides: a ÷ sin A = b ÷ sin B = c ÷ sin C. That common ratio equals the diameter of the circle through the three corners, so the circumradius is R = a ÷ (2 sin A). It works for acute, right and obtuse triangles alike (OpenStax Algebra and Trigonometry 2e, section 10.1).

When do I use the law of sines instead of the law of cosines?

Use the law of sines when you know a side together with its opposite angle: two angles and any side (ASA or AAS), or two sides and an angle opposite one of them (SSA). Use the law of cosines when you know three sides (SSS) or two sides and the angle between them (SAS). The triangle calculator on this site handles SSS, SAS and ASA; this page adds the AAS and SSA cases.

Why can SSA give two triangles?

Because sin B = sin (180° − B). The law of sines gives sin B, and the calculator’s inverse sine returns only the acute angle. The obtuse angle 180° − B has the same sine, and it makes a second valid triangle whenever the angles still add up to less than 180°. With A = 35°, a = 6 and b = 8 the two answers are B = 49.9° and B = 130.1°, with third sides 10.4 and 2.7 (OpenStax, Example 2).

How do I know whether there are 0, 1 or 2 triangles?

Compare side a with the height h = b·sin A. If A is acute: a < h gives no triangle, a = h gives one right triangle, h < a < b gives two triangles, and a ≥ b gives one. If A is 90° or more: a ≤ b gives none, and a > b gives one. The calculator prints the height so you can see which case applies.

What if sin B comes out above 1?

Then the side opposite A is too short to reach the base, and no triangle exists. For A = 50°, a = 4 and b = 10, sin B = 10 sin 50° ÷ 4 = 1.915, which is impossible because a sine never exceeds 1 (OpenStax, Example 4). Check the numbers for a typo before assuming the problem is flawed.

Which side goes with which angle?

Side a is opposite angle A, side b opposite B and side c opposite C. In the ASA case the known side lies between the two known angles, so it is opposite the third angle C; the “between” option on this page handles that. Mixing up the pairs is the most common reason an answer does not check.

Can I use radians?

This calculator takes angles in degrees. If your angle is in radians, multiply by 180 ÷ π first, or use the radians to degrees converter. A calculator left in the wrong angle mode is the second most common source of wrong triangle answers.

The method in three steps

First, if two angles are known, find the third as 180° minus their sum. Second, write one ratio that has a known side and its known opposite angle, for example a ÷ sin A. Third, set it equal to the unknown pair: b = a·sin B ÷ sin A and c = a·sin C ÷ sin A. The area is then ½·a·b·sin C. The circumradius is R = a ÷ (2 sin A), the same number for every side of the triangle. The calculator does all of this and lists the circumradius as a check: a ÷ sin A must equal 2R for all three sides.

Worked example: two angles and a side (AAS)

OpenStax Example 1: angles 50° and 30°, with the 10-unit side opposite the 50° angle. The third angle is 180° − 50° − 30° = 100°. The side opposite 30° is 10 × sin 30° ÷ sin 50° = 6.5, the side opposite 100° is 12.9, and the area is 32.1 square units. Enter 50, 30, 10 with “opposite angle A” to reproduce it.

Worked example: the ambiguous case (SSA)

OpenStax Example 2: A = 35°, a = 6, b = 8. The height is 8 × sin 35° = 4.59, and 6 lies between that height and 8, so two triangles exist. Triangle 1: B = 49.9°, C = 95.1°, c = 10.4, area 23.9. Triangle 2: B = 130.1°, C = 14.9°, c = 2.7, area 6.2. Contrast OpenStax Example 3 (relabeled here as A = 85°, a = 12, b = 9; the book calls them γ, c and b): only B = 48.3° works, because the obtuse alternative leaves a negative third angle; then C = 46.7° and c = 8.8.

Limits and common mistakes

Rounding an angle before using it in the next ratio drifts the sides: keep four or more digits until the end. In the SSA case a calculator’s inverse sine never shows you the obtuse solution, so you must test 180° − B yourself; this page does it for you and also covers the borderline: with A = 30° and b = 10 the boundary side is 5, and 90° is then the only solution (a right triangle). Measurements from the field are rarely exact, and SSA triangles with a side close to the height are extremely sensitive: a 1% change in a can swing B by many degrees. When you can choose, measure an angle you can use in an ASA or AAS setup instead.

Sources: OpenStax, Algebra and Trigonometry 2e, section 10.1 “Non-right Triangles: Law of Sines” (Examples 1 to 4 are automated tests of this calculator). Related: triangle calculator (three sides, two sides and the angle between, two angles and the side between), right triangle calculator, trigonometry calculator, radians to degrees converter and Pythagorean theorem calculator.