The area of a regular polygon with apothem a and perimeter P is?

Answer 1

Area = pa/2

Join each vertex to the centre of the n-sided regular polygon.

Then the apothem a is the height of each of the isosceles triangles thus created (as each side between a vertex and the centre must be the same length) and their base is the side s of the polygon.

Thus the area of the polygon is:

Area = n x (sa/2)

= nsa/2

But ns = the sum of all the side lengths = perimeter p. Thus:

Area = pa/2

Note: for a hexagon, the triangles created are equilateral, but an equilateral triangle is a special case of an isosceles triangle in that the base is also the same length as the other two equal sides.