What is the formula to determine the sum of the interior angles of a polygon?

The formula to determine the sum of the interior angles of a polygon is derived from the fact that a polygon can be divided into triangles. For any polygon with ( n ) sides, drawing diagonals from one vertex to the other vertices forms ( n - 2 ) triangles. Since the sum of the angles in each triangle is always 180 degrees, the total sum of the interior angles of the polygon can be calculated by multiplying the number of triangles by 180 degrees.

Therefore, the sum of the interior angles is given by the formula ( (n - 2) * 180 ). This encapsulates the idea that as the number of sides ( n ) of the polygon increases, the number of triangles formed also increases, hence increasing the sum of the interior angles.

This means that for a triangle (which has 3 sides), the formula yields ( (3 - 2) * 180 = 1 * 180 = 180 ) degrees, which is correct. For a quadrilateral (4 sides), it gives ( (4 - 2) * 180 = 2 * 180 = 360 ) degrees, aligning with our expectations. The formula consistently produces valid results for all polygons,