The Ultimate Guide to Solving Triangles

The triangle, a simple three-sided polygon, is one of the most fundamental shapes in geometry and plays a crucial role in countless fields, including engineering, physics, architecture, and navigation. To "solve a triangle" means to find the lengths of all its sides and the measures of all its angles, given a few known pieces of information. This powerful calculator is designed to be a comprehensive tool for this purpose, applying key trigonometric principles to provide a complete solution from a minimal set of data. Whether you are a student learning trigonometry, a land surveyor calculating distances, or an engineer analyzing forces, understanding how to solve triangles is an essential skill.

The Core Principles: What You Need to Know

To solve any triangle, you must start with at least three known properties, and at least one of them must be a side length. The common combinations of known values are:

• SSS (Side-Side-Side): The lengths of all three sides are known.
• SAS (Side-Angle-Side): The lengths of two sides and the measure of the angle between them are known.
• ASA (Angle-Side-Angle): The measures of two angles and the length of the side between them are known.
• AAS (Angle-Angle-Side): The measures of two angles and the length of a side that is not between them are known.

The infamous **SSA (Side-Side-Angle)** case, where two sides and a non-included angle are known, is called the "ambiguous case" because it can sometimes result in two different valid triangles, one triangle, or no triangle at all. This calculator attempts to find a valid solution but will warn you if one cannot be found from the given inputs.

The Mathematical Toolkit: Laws of Sines and Cosines

The two most powerful tools in trigonometry for solving general (non-right-angled) triangles are the Law of Sines and the Law of Cosines.

1. The Law of Sines

The Law of Sines establishes a relationship between the sides of a triangle and the sines of their opposite angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

The Formula: a / sin(A) = b / sin(B) = c / sin(C)

When to Use It: The Law of Sines is used when you know either two angles and one side (ASA or AAS) or two sides and a non-included angle (the ambiguous SSA case).

2. The Law of Cosines

The Law of Cosines relates the lengths of the three sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem.

The Formulas:

• c² = a² + b² - 2ab * cos(C)

This can be rearranged to solve for an angle if you know all three sides, for example: A = arccos((b² + c² - a²) / 2bc)

When to Use It: The Law of Cosines is used when you know either all three sides (SSS) or two sides and their included angle (SAS).