Solution of triangles

Overview of the particular steps when solving a triangle

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. A triangle can be solved when given the any of the following information:

• Three sides (SSS)
• Two sides and the included angle (SAS)
• Two sides and an angle not included between them (SSA)
• A side and the two angles adjacent to it (ASA)
• A side, the angle opposite to it and an angle adjacent to it (AAS)

Solving a triangle

Three sides given (SSS)

If one knows the lengths of the three sides, one can:

1. Find the first angle by using the law of cosines, which states that $c^{2}=a^{2}+b^{2}-2ab(\cos x)\,$ where a, b and c are the sides of the triangle and x is the angle opposite to c
2. Find the second and third angle using the much easier law of sines, which states that ${\frac {a}{\sin x}}={\frac {b}{\sin y}}={\frac {c}{\sin z}}\,$ where x, y, z are the angles opposite to sides a, b and c respectively. The third angle can also be found using the angle sum property of a triangle, which states that the sum of the angles of a triangle equals 180°.

Two sides and the included angle given (SAS)

If one knows the lengths of two of the sides and the angle between them, one can:

1. Find the length of the third side by using the law of cosines.
2. Find the second and third angle using law of sines. The third angle can also be found using the angle sum proprety of a triangle, which states that the sum of the angles of a triangle equals 180°.

Two sides and non-included angle given (SSA)

If one knows the lengths of two sides and an angle not included by the two sides, one can:

1. Find the second angle by using the law of sines.
2. Find the third angle by using the angle sum property a triangle to find the third angle.
3. Use the law of sines again for finding the third side.

Solving SSA triangles: The ambiguous case

In some cases an SSA triangle can have two possible solutions, or even no possible solution.

The followng conditions can be used to determine how many solutions are possible with an SSA triangle: (x is the given angle, a is the side opposite to x, and b is the other given side.)

• If x < 90°:
  • If a ≥ b, only 1 solution is possible.
  • If a < b:
    • If a > b * sin x, 2 solutions are possible.
    • If a = b * sin x, only 1 solution is possible.
    • If a < b * sin x, no solution is possible.
• If x > 90°:
  • If a ≤ b, no solution is possible.
  • If a > b, only 1 solution is possible.

A side and two adjacent angles given (ASA)

If one knows the length of one side and the two angles adjacent to it, one can:

1. Find the third angle by using the angle sum property of a triangle.
2. Find the lengths of the other two sides by using the law of sines.

A side, one adjacent angle and the opposite angle given (AAS)

The procedure for solving an AAS triangle is same as that of an ASA triangle: First, find the third angle by using the angle sum property of a triangle, then find the other two sides using the law of sines.

Checking the solutions

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

Solving spherical triangles

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

See also

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

External links

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