Question

How do you find the apothem of an "n-gon" (specifically a hexagon) if you're only given the radius? The exact question is:
The radius of regular hexagonal sandbox is 5 ft. What is the area to the nearest square foot?
Thanks!x

Answer 1

I'm not sure if there's a general formula you can apply directly, or if you can just solve this by reasoning through it. Chances are that you can derive the general formula with the reasoning.

In any case, you can split up the area of a regular \(n\)-gon into \(n\) isosceles triangles, which can in turn be split in half into identical right triangles. The \(n\)-gon's radius is the hypotenuse of each triangle, and the apothem is altitude of each isosceles sub-triangle.

Consider a square (a regular 4-gon) as a simple example:

Let's say the radius is \(\sqrt2\), for later simplicity. The approach I want to describe involves finding the interior angle of each vertex. Since we have a square, each interior angle is \(90^\circ\). Each isosceles sub-triangle bisects this angle, so each right triangle within the sub-triangles is \(45-45-90\). This tells you that, if \(r\) is the radius and \(a\) is the apothem, then \(a=1\).

For the general \(n\)-gon with given radius \(r\), you'll want to find the interior angle \(x\) of the \(n\)-gon, then using trigonometry, you can determine the apothem \(a\).

Solve for \(a\) in the above equation. You'll then have everything you need to find the area of each sub-triangle, and thus the area of the \(n\)-gon.