Triangle Calculator: Sides, Angles, Area

How do you solve a triangle?

A triangle has three sides and three angles. When three of them are known, and at least one is a side, the other three can be calculated. Which theorem applies depends on what is known.

• Three sides (SSS): the angles follow from the law of cosines.
• Two sides and the included angle (SAS): the third side follows from the law of cosines.
• Two angles and a side (ASA, AAS): the third angle is what remains of 180 degrees, and the sides follow from the law of sines.
• Two sides and an angle opposite one of them (SSA): the law of sines applies and two different triangles may result.

Formulas

Law of cosines: a² = b² + c² − 2 × b × c × cos A. Law of sines: a / sin A = b / sin B = c / sin C. Area: b × c × sin A / 2. With three sides the area also follows from Heron's formula.

Example

For sides 3, 4 and 5, cos C = (9 + 16 − 25) / 24 = 0, so C = 90 degrees. The other angles are 36.87 and 53.13 degrees and the area is 6.

The ambiguous case

Given a = 6, b = 10 and A = 30 degrees, sin B = 10 × 0.5 / 6 = 0.833. Both 56.44 and 123.56 degrees have this sine and both make a valid triangle: the third side is 11.98 in one and 5.34 in the other. The calculator draws both. If side a is shorter than b × sin A there is no triangle, and if it is longer than b there is exactly one.

Limits

Any two sides must add up to more than the third, and two angles must add up to less than 180 degrees. The calculator covers plane triangles: spherical triangles and long distances on the earth's surface are out of scope.