Law of Sines Calculator
With known side a 10, angle a (opposite side a) 30 °, angle b 45 °, law of sines comes to 14.1421 — side b. It is reached in 6 steps, the last of which is 20 * sin(45 * pi / 180), and each one is printed on the page with its numbers filled in. The formula is the one published by Encyclopedia of Mathematics, not an approximation fitted to it.
Solves a triangle from one side and two angles: the third angle, the remaining two sides and the area, using a/sin A = b/sin B.
Formula and sources checked · How we check
Known side a 10, Angle A (opposite side a) 30 °, Angle B 45 °
14.1421
Side b for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Angle C
180 - 30 - 45105 °- a / sin A
10 / sin(30 * pi / 180)20- Side b
20 * sin(45 * pi / 180)14.142- Side c
20 * sin(105 * pi / 180)19.319- Perimeter
10 + 14.142136 + 19.31851743.461- Area
0.5 * 10 * 14.142136 * sin(105 * pi / 180)68.301
Worked example
Angle C is 180 − 30 − 45 = 105 degrees. The ratio a/sin A is 10/0.5 = 20, so side b is 20 × sin 45° = 14.1421 and side c is 20 × sin 105° = 19.3185. The area is 68.3013 square units. Notice the largest side sits opposite the largest angle, which is the check worth making on every triangle.
How to work it out yourself
- 1.This wants one side and the two angles you know. If instead you have two sides and the angle between them, the law of cosines is the one that works.
- 2.The three angles must add to 180 degrees. Enter two that already sum to 180 or more and there is no triangle to solve.
- 3.Check the answer by eye: the longest side is always opposite the largest angle, and the shortest opposite the smallest.
The formula
- Angle C
180 - 30 - 45 - a / sin A
10 / sin(30 * pi / 180) - Side b
20 * sin(45 * pi / 180) - Side c
20 * sin(105 * pi / 180) - Perimeter
10 + 14.142136 + 19.318517 - Area
0.5 * 10 * 14.142136 * sin(105 * pi / 180)
Source: Encyclopedia of Mathematics — sine theorem, NIST DLMF §4.14 — trigonometric functions
Questions people actually ask
- What is the ambiguous case?
- It arises when you know two sides and an angle that is not between them (SSA). The sine rule can then give two different triangles, because sin θ and sin(180° − θ) are equal. This calculator takes two angles and one side instead, which is never ambiguous.
- When do I use the sine rule rather than the cosine rule?
- The sine rule needs a matched pair — a side and the angle opposite it. If you have that, use it. If you have two sides and the angle between them, or all three sides, only the cosine rule will start you off.
- What is the ratio a/sin A actually equal to?
- The diameter of the triangle’s circumscribed circle. In the worked example it is 20, so the circle through all three vertices has a radius of 10 — which is also why the ratio is the same for all three sides.
Related
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