Law of Sines Calculator
- The law of sines says a ÷ sin A = b ÷ sin B = c ÷ sin C.
- Find the third angle: C = 180° − 40° − 60° = 80°.
- Use the known side to get the ratio: a ÷ sin A = 10 ÷ sin 40° = 15.5572.
- Side b = 15.5572 × sin 60° = 13.473.
- Side c = 15.5572 × sin 80° = 15.3209.
- Angle sum: the three angles of a triangle always add up to 180°.
- Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C, so each side is proportional to the sine of its opposite angle.
- a ÷ sin A = 15.5572, b ÷ sin B = 15.5572, c ÷ sin C = 15.5572. The three ratios match.
- The angles add up: 40° + 60° + 80° = 180°.
- Triangle inequality: the two shorter sides add up to 23.473, which is more than the longest side, 15.3209.
Law of Sines Calculator: Solve a Triangle from Any Three Values
This free law of sines calculator finds the missing sides and angles of a triangle. Enter any three values, with at least one side, and it fills in the rest, draws the triangle to scale, and shows every step. It covers the cases the law of sines is used for and also the ones that start with the law of cosines, and it warns you when no triangle fits or when two do.
Quick answer: if angle A is 40°, angle B is 60° and side a is 10, then angle C is 80°, side b is about 13.47 and side c is about 15.32.
What Is the Law of Sines?
The law of sines, also called the sine rule, links each side of a triangle to the angle opposite it. If the sides are a, b and c, and the angles opposite them are A, B and C, then a ÷ sin A = b ÷ sin B = c ÷ sin C. Because the three ratios are equal, one known pair of a side and its opposite angle lets you find every other side from its angle, or every other angle from its side. Unlike the Pythagorean theorem, it is not limited to right triangles, though the Pythagorean theorem calculator is the quicker tool when there is a right angle.
Which Cases Can Be Solved?
The method depends on which three values you know. The calculator picks it automatically.
| Known values | Name | Method |
|---|---|---|
| Two angles and a side between them | ASA | Third angle, then law of sines |
| Two angles and a side not between them | AAS | Third angle, then law of sines |
| Two sides and an angle opposite one | SSA | Law of sines, check for two triangles |
| Two sides and the angle between them | SAS | Law of cosines, then law of sines |
| Three sides | SSS | Law of cosines |
| Three angles | AAA | Shape only, size cannot be found |
The Ambiguous Case (SSA)
When you know two sides and an angle that is opposite one of them, the numbers do not always describe a single triangle. Sine gives the same value for an angle and for 180° minus that angle, so the second angle can be acute or obtuse. If both choices leave room for a third angle, there are two triangles, and the calculator shows each one with a switch. If the known side opposite the angle is too short to reach, there is no triangle at all. If it is as long as the other known side or longer, there is exactly one.
Law of Sines or Law of Cosines?
The law of sines needs a side and the angle opposite it to get started. With two angles you can always find the third, so those cases work directly. With two sides and the angle between them, or with all three sides, there is no such pair yet, so the law of cosines comes first. After one new side or angle is found, the law of sines finishes the job. The step-by-step box shows which rule was used at each stage.
Using Triangle Results
Besides the missing values, the calculator gives the perimeter, the area, the circumradius and the type of triangle, such as acute scalene or right isosceles. Solving triangles is used in surveying, navigation, building and design, wherever a distance cannot be measured directly. For the angles themselves, the sin cos tan calculator shows exact values, and the scientific calculator and the math solver help with the arithmetic. To work with shapes built on triangles, try the area calculator and the circle calculator.
How to Use This Law of Sines Calculator
Fill in three values
Type any three of the six boxes, with at least one side. Side a is opposite angle A, and so on.
Read the solution
The empty boxes show the calculated values, and the results appear beside them.
Check for two triangles
If two fit, use the Triangle 1 and Triangle 2 buttons to switch.
Follow the working
See each step, the rules used and the checks on the answer.