Any regular polygon inscribed in a circle- Math Central

Math Central

Quandaries & Queries

Question from DJ, a parent:

Circle with r=12" is inscribed in a regular octagon. What is the length of each octagon segment?

Hi DJ.

Here's a method that solves this problem for any regular n-gon inscribed in a circle of radius r.

A regular n-gon divides the circle into n pieces, so the central angle of the triangle I've drawn is a full circle divided by n: 360°/n.

The Law of Cosines applies to any triangle and relates the three side lengths and a single angle, just as we have here. In general it says this:

The Law of Cosines:
For any triangle with side lengths a, b and c,
c² = a² + b² - 2ab cos C
where angle C is the angle opposite side length c.

This is the key to the answer.

In the case of our regular n-gon, we have a = b = r and C = 360°/n, so we simplify it to:

c² = r² + r² - 2rr cos (360°/n)
c² = 2r² - 2r² cos (360°/n)
c² = r² (2 - 2 cos (360°/n))

which makes a final equation:

If you wanted the perimeter of the n-gon, you would just multiply this by n of course.

For your question DJ, just put 12" in for r and 8 in for n and you can calculate c.

Hope this helps,
Stephen La Rocque

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.