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
(r cos π/n) (r sin π/n)
= r2 cos π/n sin π/n
= (r2/2) sin(2π/n)
and the area of the entire polygon is
A = n (r2/2) sin(2π/n) = πr2 (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 πr2, just like we expected.
Now the area of the circle with radius r is πr2, so the difference between the area of the circle and the area of the polygon is
πr2 – πr2 (n/2π) sin(2π/n)
= πr2 (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.