When all three sides are given but no angles, which rule is appropriate?

When all three sides of a triangle are known, the appropriate rule to use is the Cosine rule. This rule allows us to find the angles of the triangle or confirm the relationship between the sides. The Cosine rule states that for any triangle with sides a, b, and c, and the angle opposite to side c as C, the relationship can be expressed as:

c² = a² + b² - 2ab * cos(C).

This formula is particularly useful when you need to determine an angle when the lengths of all three sides are provided. It provides a means to relate the sides of the triangle directly to each other through cosine of the angles.

In contrast, the Sine rule is utilized when you have a pair of angles and one side, or a pair of sides and a non-included angle. The Pythagorean theorem specifically applies to right-angled triangles, while the Area rule is not a standard rule for finding missing information based solely on the three sides. Thus, the Cosine rule is uniquely suited for scenarios involving three known side lengths.