How is the area of a regular polygon calculated using the apothem?

The area of a regular polygon can be calculated using its apothem with the formula that incorporates both the apothem and the perimeter. Specifically, the correct formula is A = ½ * a * p, where 'A' represents the area, 'a' is the apothem, and 'p' is the perimeter of the polygon.

The reason this formula is valid is that the apothem acts as a height when the polygon is divided into triangles that have one vertex at the center of the polygon and the base along the sides. The area of each of these triangles can be found using the base of the triangle (which corresponds to the segment of the side of the polygon) and the height, which is the apothem. Since a regular polygon can be perceived as being made up of several such triangles, summing their areas leads back to the formula that uses the apothem multiplied by half the perimeter.

This formula effectively captures both the dimensions of the polygon and its regular shape, providing a straightforward method to find the area given the necessary length measurements. Using this approach, the relationship between the apothem and the perimeter ensures that you account for the entire surface area of the polygon in your calculations.