Finding the radius of a regular polygon given area?

Question

How would you solve for the radius of a regular polygon given its area? This is part of a large problem: Find one complex number whose 12th roots connected results in a polygon with area 6 sq units.

Roots increase around a circle in increments of π/6, but to find it in rcis(θ) I would need to find the radius of the regular dodecagon the roots create. You can find all three angles of the triangles inside the polygon, as well as the area, but that leaves you with too many variables to solve for. Am I approaching this problem the wrong way, or am I missing something. Thanks!

Answer 1

Let me tackle the first part first.

The interior angle of a regular polygon of n sides is (n-2)180/n.

Let θ = 1/2 of the interior angle.

The apothem of the polygon is (s tan θ)/2 where s is the side of the polygon.

The area of the polygon then is (ns² tan θ)/4

Therefore, from the area you can calculate the side.

The radius = sqrt[apothem² + (1/4)s²].

This is not a very satisfying expression but I think it correctly answers your question.

Of course, if you look at a figure you may see that there are other ways to write this last expression.

Of one thing I am pretty certain: in order to get from area to radius, you need to calculate the apothem.

Your second paragraph is sufficiently unclear that I am not sure what it is you want to do.

If you can clarify what you want to do specifically, I will try to help you further.