By apothem and perimeter

The formula for the area of a regular polygon given its apothem (the distance from the center of the polygon to the midpoint of a side) and perimeter (the total length of all sides) is:

Area = 0.5 x apothem x perimeter

Explanation

For a regular polygon, the apothem is the distance from the center of the polygon to the midpoint of one of its sides. The perimeter is the sum of the lengths of all the sides of the polygon. To find the area of a regular polygon given its apothem and perimeter, you can use the above formula, which states that the area is equal to half of the product of the apothem and the perimeter.

This formula works because if you draw the apothem of a regular polygon, it will divide the polygon into congruent triangles, each with a base equal to one side of the polygon and a height equal to the apothem. The area of each triangle is half the product of the base and height. Since there are n such triangles (where n is the number of sides of the polygon), the total area of the polygon is n times the area of one triangle, or:

Area = n x 0.5 x base x height

The base of each triangle is one side of the polygon, so the total length of the bases is equal to the perimeter. The height of each triangle is the apothem, so we can substitute these values in the equation and simplify to get:

Area = 0.5 x apothem x perimeter