Write an explanation that justifies this statement: the area of a regular polygon is half the product of the apothem and the perimeter

Take for example a hexagon.

The area of the hexagon is given by finding the area of one of the inner triangles formed using the center of the hexagon, one side of the hexagon and the apothem as the height. 

The area of a triangle is A=1/2bh; in this case, h is the apothem and b is the side length.  

In our hexagon, there will be 6 of these triangles:
A = 6(1/2)(s)(a)

6s is also an expression for the perimeter of the hexagon, because it is a regular hexagon.

Thus our formula for area is the same as saying the perimeter of the hexagon, 6s, multiplied by 1/2 and by the apothem.