# Approximating a circle with a polygon

A circle is the limiting case of a polygon with lots of sides, so a reasonable question to ask (like I was recently asked) is exactly how many sides a polygon has to have for its area to be a good approximation to the area of a circle. Here's my answer to this question.

Suppose that we have a regular polygon with *n* sides and that the distance from the center of the polygon to any of its vertices is *r*. If we look at the wedge formed by drawing lines from the edges of a single side of the polygon to the center, we get something that looks like this, where the angle in the center of the wedge is 2π/*n*.

If we divide this wedge into two right triangles, we can then use some trigonometry to find the lengths of the sides of each of the triangles in terms of the length *r* and the angle 2π/*n*. This gives us something like this:

This means that the area of each of the wedges is

(

rcos π/n) (rsin π/n)=

r^{2}cos π/nsin π/n= (

r^{2}/2) sin(2π/n)

and the area of the entire polygon is

A=n(r^{2}/2) sin(2π/n) = πr^{2}(n/2π) sin(2π/n)

Now what happens we increase the number of sides of the polygon?

As *n* gets big, 2π/*n* gets close to 0 so that

(

n/2π) sin(2π/n) = sin(2π/n)/(2π/n)

gets close to 1, so we have that *A* gets close to π*r*^{2}, just like we expected.

Now the area of the circle with radius *r* is π*r*^{2}, so the difference between the area of the circle and the area of the polygon is

π

r^{2 }– πr^{2}(n/2π) sin(2π/n)= π

r^{2}(1 – (n/2π) sin(2π/n))

If we plot

f(n) = 1 – (n/2π) sin(2π/n)

we find that it looks like this:

so it’s clearly possible to get a good approximation with not too many sides.

We actually have that *f*(8) = 0.900316, so that using just 8 sides gives us less than 10 percent error. To get to 5 percent error it turns out that we need to use 12 sides and to get 1 percent error we need to use 26 sides. This means that it's probably reasonable to say that a 26-sided polygon (icosikaihexagon?) is a good approximation to a circle. Here's a 26-gon that I drew using Google Sketchup that seems to show that a 26-gon is fairly circle-like:

Polygons with fewer sides might also be OK, depending on exactly how good you want your approximation of a circle to be.

## Radius to area of a circle

It is usual to use the formula to find the area of a circle using its radius, but sometimes kids ask where that formula came from or is there any other method to find the area of a circle?

I think above is the best example to approximate the area of a circle and easy explanation for kids to understand the concept.

Thanks

Manjit.

Reply ↓