Which formula is used to find the area of a triangle given two sides and the included angle?

The formula to find the area of a triangle when two sides and the included angle are known is given by A = ½ ab sin C. In this formula, 'a' and 'b' represent the lengths of the two sides, and 'C' represents the measure of the included angle between those two sides.

This approach works because the area of a triangle can be thought of as half the product of the base and height. In this case, the two sides act as the legs of a right-angled triangle formed by dropping a height from the vertex opposite the included angle down to the line segment forming the base. The sine of the angle provides the necessary relationship to find the height with respect to one of the sides, resulting in the area being calculated correctly.

This formula is particularly useful in situations where you do not have the height directly available, but you do have the lengths of two sides and the angle between them, allowing for versatile applications in various geometric problems.