How do you find the perimeter of a regular polygon when you only have the area and apothem of that polygon?

Answer 1

the equation for the area of a regular polygon:
1/2(Apothem x Perimeter)=Area.
Just plug in the numbers you know and solve.

Answer 2

The area of any regular polygon is the sum of the area of n triangles, each having area A = (1/2)(b)(h), where b is the total perimeter divided by the number of triangles in the polygon, which is n, and h is the apothem of the polygon. Substituting (p/n) in for b and a in for h, we get this equation:

A = (1/2)(p/n)(a).

Since there are n triangles, the total area is A' = n[(1/2)(p/n)(a)] = (1/2)(p)(a). Now all we have to do is manipulate this equation algebraically to find p:

A' = (1/2)(p)(a)
2A' = pa
2(A'/a) = p.

So the perimeter of any regular polygon is 2 times the ratio of the total area of the polygon to its apothem.