Franklin’s Blog

Area of a Regular Polygon

2016 Nov 8

Derive a formula for the area of a regular polygon, given the side length and number of sides.

Let a regular polygon with n sides have side length l and apothem a. By a general formula, the area of this polygon is given by
\[A=\frac{1}{2}lan\] However, when finding the area of such a polygon, it can be tiresom to solve for the apothem given the side length and the number of sides, so we can make a general formula. If a cut is made from each vertex of the polygon to the center, it will be cut up into \(n\) congruent isoscles triangles looking like this:

Fig 1

We can now cut each of these triangles in half with an altitute and end up with \(2n\) right triangles that look like this:

Fig 2

Now that we have a right triangle, we can use right triangle trigonometry to say that

\[tan(\frac{360}{2n})=\frac{l}{2a}\]
\[2a=\frac{l}{tan(\frac{180}{n})}\]
\[a=\frac{l}{2tan(\frac{180}{n})}\]

When we plug this into our original formula, we get

\[A=\frac{1}{2}lan\]
\[A=\frac{1}{2}l(\frac{l}{2tan(\frac{180}{n})})n\]
\[A=\frac{l^2n}{4tan(\frac{180}{n})}\]

Which gives us a formula for the area of a regular polygon given only the side length and the number of sides.