Law of sines calculator
Angles of 40° and 60° with a = 10 opposite the 40° give b = 13.473 and c = 15.3209 in the law of sines calculator, since a / sin A = 15.5572.
Missing sides
- Missing sides, by the law of sines C = 180° - 40° - 60° = 80°. Then a / sin A = 10 / 0.6428 = 15.5572, so b = 15.5572 × sin 60° = 13.473 and c = 15.5572 × sin 80° = 15.3209.
- b = 13.473, c = 15.3209
- Sides
- a = 10, b = 13.473, c = 15.3209
- Angles Summing to 180 degrees, as every plane triangle must.
- A = 40deg, B = 60deg, C = 80deg
- Area
- 66.3414
- Perimeter
- 38.7939
- Type
- scalene, acute
- Heights The perpendicular from each side to the opposite vertex. Area is half the base times the height for any of the three.
- to a: 13.2683, to b: 9.8481, to c: 8.6603
- Circumradius The radius of the circle through all three vertices.
- 7.7786
- Inradius The radius of the largest circle that fits inside.
- 3.4202
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 30-40-50 |
| 5 | 12 | 13 | 10-24-26, 15-36-39 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
| 20 | 21 | 29 | 40-42-58 |
| 9 | 40 | 41 | 18-80-82 |
- The Pythagorean theorem holds only for right triangles. For any other triangle the law of cosines generalizes it: c squared equals a squared plus b squared minus 2ab cos(C), which reduces to Pythagoras when C is 90 degrees and cos(C) is zero.
- Any two sides must add to more than the third, or the shape does not close. That triangle inequality is checked here before anything else is computed.
How the Triangle Calculator works
The law of sines says that in any triangle, each side divided by the sine of the angle opposite it gives the same number: a / sin A = b / sin B = c / sin C. The law of sines formula solves a triangle whenever one side and its opposite angle are known, plus one more side or angle.
Worked example, two angles and a side not between them (AAS): A = 40°, B = 60° and a = 10. The third angle is C = 180° - 40° - 60° = 80°. The ratio is a / sin A = 10 ÷ 0.6428 = 15.5572, so b = 15.5572 × sin 60° = 13.473 and c = 15.5572 × sin 80° = 15.3209. The area is 66.3414.
With two angles and the side between them (ASA), find the third angle first, and the given side and its opposite angle become the known pair. For A = 30°, B = 45° and c = 10: C = 105°, c / sin C = 10.3528, a = 5.1764 and b = 7.3205.
Two sides and an angle opposite one of them (SSA) is the ambiguous case. Solving sin B = b sin A ÷ a gives a sine, and two angles under 180° share every sine: B and 180° - B. Take a = 7, b = 10 and A = 30°. Then sin B = 10 × 0.5 ÷ 7 = 0.7143, so B is 45.58° or 134.42°. Both leave room for a third angle, so two different triangles fit: c = 13.5592 with C = 104.42°, or c = 3.7613 with C = 15.58°.
The count depends on the height h = b sin A, which is 5 here. If a is shorter than h there is no triangle; equal to h, one right triangle; between h and b, two; at least b, one. When A is 90° or more there is one triangle if a is longer than b and none otherwise.
Two sides with the angle between them, or three sides, need the law of cosines instead.
Triangle Calculator FAQ
Why is a / sin A the diameter of the circumcircle?
Draw the circle through the three vertices and the diameter from B. The angle at its far end equals A, because both stand on the same chord a, and the angle in a semicircle is 90°, so sin A = a ÷ diameter. In the 40-60-80 example the circumradius is 7.7786, and 2 × 7.7786 = 15.5572.
Can the law of sines find an angle?
Yes, from two sides and the angle opposite one of them: sin B = b sin A ÷ a. With a = 12, b = 10 and A = 30°, sin B = 0.4167 and B = 24.62°. The other angle with that sine, 155.38°, would leave no room for C, so there is one triangle.
What if sin B comes out more than 1?
Then no triangle has those measurements: side a is too short to reach the line it has to meet. With a = 4, b = 10 and A = 30°, sin B = 1.25, and a is shorter than the height b sin A = 5.
More from the Triangle Calculator
- Right triangle calculator
- Law of cosines calculator
- Equilateral triangle calculator
- Isosceles triangle solver
- Pythagorean theorem calculator
- Triangle area calculator