why does pa/2 work for polygons

Question

Why does the formula apothem times perimeter divided by two work as a way to find the area of a regular polygon?

Answer 1

connect the center of the polygon to each vertex, there are n congruent triangles.

the apothems drawn perpendicular to each side, or bases of the triangles, is the height of each triangle

each triangles area = side x apothem /2 and all the triangles areas =n(sidexapothem/2)

n x side = perimeter or ns=p, so the area of the polygon is pa/2