Law of sines calculator

A law of sines calculator solves a triangle that is not right-angled from two angles and a side, or from two sides and an angle opposite one of them. The law of sines says each side divided by the sine of its opposite angle gives the same number: a/sin A = b/sin B = c/sin C. Type your three values into our Trigonometry Calculator and it finds the rest, including both triangles when two fit.

There is no separate law of sines mode to switch to. The calculator looks at which three values you typed, picks the law of sines when it is the right tool, and says so in the method line under the buttons.

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Trigonometry Calculator

Interactive triangle solver with step-by-step

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How to use the law of sines calculator

1. Open the calculator and choose Standard triangle at the top of the sidebar. It starts on the unit circle.
2. Under Triangle, find the panel "Type any three values, including a side, then Solve".
3. Type your values in the a, b, c and A, B, C boxes. Side a is opposite angle A, and so on, so match each side to the angle across from it.
4. Press Solve. The triangle is redrawn, and Measurements lists every side and angle, the area and the perimeter.
5. If two triangles fit, buttons for Triangle 1 and Triangle 2 appear. Press either one to draw it.

The method line tells you which case you had, such as "Two angles and a side (AAS): the third angle, then the sine rule". The Formulas section shows the law of sines with the common ratio for your triangle. The calculator gives the results and the method but not the working, so the examples below show each step. Angles are in degrees unless you switch Degrees off under Display.

The law of sines formula

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

You only ever use two parts at once: one complete pair, a side with its opposite angle, and one more value. In the UK the same rule is called the sine rule, and our sine rule guide covers its proof and exam use.

When to use the law of sines

If you do not have a side with its opposite angle, the law of sines cannot start, and you need the law of cosines instead.

Worked examples you can check in the calculator

Example 1: two angles and the side between them (ASA)

In a triangle, A = 40°, B = 60° and c = 10. Find the other sides. First the third angle: C = 180° − 40° − 60° = 80°. Now c and C are a complete pair, so:

Type A = 40, B = 60 and c = 10, then Solve. The method line says ASA, and Measurements gives a = 6.53, b = 8.79 and C = 80°.

Example 2: two angles and another side (AAS)

B = 32°, C = 105° and b = 14. The third angle is A = 180° − 32° − 105° = 43°, and b with B is the complete pair:

The longest side, c, is opposite the largest angle, 105°, which is a good check. Try one in the course now, then press Mark:

Example 3: finding an angle, with one answer

a = 10, b = 6 and A = 70°. Find angle B. Use the angles-on-top form:

Every sine value between 0 and 1 also belongs to an obtuse angle, here 180° − 34.32° = 145.68°. But 145.68° + 70° is more than 180°, so it cannot be in the triangle. There is one triangle: B = 34.32°, C = 75.68° and c = 10.31.

Law of sines ambiguous case calculator

With two sides and an angle opposite one of them (SSA), there can be two triangles, one or none. This is the ambiguous case, and it is where a calculator that only gives one answer can lead you wrong.

Example 4: two triangles fit

a = 6, b = 8 and A = 40°. Find B.

So B = 58.99°, or B = 180° − 58.99° = 121.01°. Both work, because 121.01° + 40° is still less than 180°. Side a can swing to meet the base in two places:

Type a = 6, b = 8 and A = 40, then Solve. The calculator draws Triangle 1 and shows a button for each: "Triangle 1: A 40.0°, B 59.0°, C 81.0°" and "Triangle 2: A 40.0°, B 121.0°, C 19.0°". If a question does not say which triangle, give both.

How many triangles? A quick test

When the given angle A is acute, compare side a with the height b sin A:

• a less than b sin A: no triangle. Side a is too short to reach.
• a equal to b sin A: one right triangle.
• a between b sin A and b: two triangles.
• a at least b: one triangle.

In Example 4, b sin A = 8 sin 40° = 5.14, and 6 lies between 5.14 and 8, so there are two. Change a to 3 and the calculator says "No triangle has these values: side a is too short to reach the third side". When A is obtuse, there is one triangle if a is longer than b, and none otherwise.

Area of a triangle with sine

Once a triangle is solved, its area is half of two sides times the sine of the angle between them: Area = ½ab sin C. For Triangle 1 in Example 4, that is ½ × 6 × 8 × sin 81.01° = 23.71. Measurements gives the area straight away, and the Formulas section shows the same formula as Area (SAS).

Common mistakes

• Pairing a side with the wrong angle. Each side goes with the angle opposite it, never an angle next to it.
• Missing the second triangle. In the SSA case, always check 180° minus your angle as well.
• Using the law of sines with SAS or SSS. With no complete side and angle pair, it cannot start. Use the law of cosines.
• A calculator in radians. sin 40 in radians is about 0.75, not 0.64. Check for DEG on the screen.
• Forgetting the angles add to 180°. Find the third angle first in ASA and AAS questions. It is often the one you need.
• Rounding the angle before finding a side. Keep the full value of sin B in your calculator for the next step.

Practise the law of sines in a short timed paper

The law of sines comes up in Geometry, Algebra 2 & SAT Math and AP Precalculus practice, and as the sine rule in GCSE Maths Higher in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. For right triangles, the right triangle calculator guide is quicker.

Read next

Maths · TrigonometryLaw of cosines calculator: SAS and SSS trianglesFree law of cosines calculator: find the third side from two sides and the angle between them, or every angle from three sides, with worked examples.