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Solving the Triangle

What does 'solving the triangle' mean?

It means that if we are given some facts about a triangle, we can find some or all of the rest. For example, if we know two sides of a right triangle we can find (or 'solve for') the third side using Pythagoras' Theorem.

To completely solve a triangle it usually means finding everything about it - all three sides and all three angles. But often we are just interested in one unknown aspect of the triangle.

Tools for solving triangles

Tool Given You can find
Interior angles sum to 180° Any two interior angles The third angle
Pythagoras' Theorem Two sides of a right triangle The third side

If you are familiar with trigonometry, the following tools can be used with any triangle:

Tool Given You can find
Law of Cosines Two sides and the angle between them The third side
Law of Sines One side and its opposite angle, and one other item Everything else

Again, if you are familiar with trigonometry, the following tools can be used with right triangles:

Tool Usage
Sine These trigonometry functions relate an angle and two other sides. If you know two of the three, you can find the other. It is beyond the scope of the this volume on Geometry to define them here. See the Trigonometry Overview for a brief description.
Cosine
Tangent

The tools listed above can be used in any combination. There is no 'right way' of solving a triangle (or any other geometric problem). Just as in woodworking, you can use the tools in many ways and in many sequences. Think about what you have been given to start, and find tools that lead you in the direction you want to go.

Other triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises