Solution of triangles

In trigonometry, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle when given some, but not all of that information. In particular:

• If one knows two of the angles one can find the third by using the fact that the sum of the three must always be 180°.
• (SSS) If one knows the lengths of the three sides, one can find the first angle by using the law of cosines, and proceed then with the much easier law of sines.
• (SAS) If one knows the lengths of two of the sides and the angle between them, one can find the length of the third side by using the law of cosines, and proceed then with the much easier law of sines.
• (SSA) If one knows the lengths of two sides and an angle between one of those and the third side, one can find the second angle by using the law of sines, and then proceed by using the sum of angles within a triangle to find the third angle, finally using the law of sines again for finding the remaining side lengths, in some cases up to a choice between two possible solutions.
• (AAS) & (ASA) If one knows the length of one side and at least two of the angles, one can find the third angle by using the sum of angles within a triangle, and the lengths of the other two sides by using the law of sines.

To check the solutions, Mollweide's formula can be used. In some cases, the law of tangents may also be useful.

When solving spherical triangles, the half-side formulae are used.

See also

• Hinge theorem
• Hansen's problem
• Snellius–Pothenot problem

Weblinks

• Triangulator - Triangle solver. Solve any triangle problem with the minimum of input data. Drawing of the solved triangle.