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If I have $2\pi/n$, where $n$ is the number of sides of a polygon, does the answer give me a length of sides necessary for me to draw the polygon with those $n$ sides as a regular polygon inside the circle?

I know it works or I believe it does. But analytically, without anything but the drawing, how can I know this is true? I am having trouble visualizing it. I want to know why it works.

Aweygan
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johnny
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  • @Aweygan I do not understand. Number is the number of sides of a polygon. The result is supposed to be a regular polygon that I can fit in a circle. It is assuming I want to get a regular polygon. – johnny Jul 26 '16 at 22:02
  • If someone is going to downvote me please help me understand how to better ask it. – johnny Jul 26 '16 at 22:03
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    @johnny When you divide $2\pi$ by the number of sides (or any other number), you get a number, like 0.314. Your first question is asking "why does 0.314 give me a regular polygon?" The answer is it doesn't: a number by itself does not give you a polygon. You need to describe what you are doing with that number to obtain a polygon from it. – Erick Wong Jul 26 '16 at 22:06
  • @Aweygan I changed it. – johnny Jul 26 '16 at 22:07
  • @ErickWong I want to draw a polygon inside a circle that is a regular polygon. I read somewhere that you can get the correct length of the sides for that polygon by dividing the number of sides of that polygon by 2*pi. – johnny Jul 26 '16 at 22:09
  • http://math.stackexchange.com/questions/720935/historic-proof-of-the-area-of-a-circle/1678093#1678093 –  Jul 26 '16 at 22:09
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    @johnny Thanks, that clarifies a lot. One thing still missing is how large your circle is. Clearly the size of the circle determines the length of the side. And is it the straight-line length you are measuring, or the arc length? – Erick Wong Jul 26 '16 at 22:31

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What you can do is divide the angle at the center of the circle by $n$, which gives angles of $\frac {2 \pi}n$ radians at the center. Now draw radii from the center to the circle at this spacing and make the points where the radii hit the circle. As all the central angles are the same, the lengths of the sides between the radii are all the same. The angles between the sides of your polygon are twice the base angle of the isoceles triangles formed by the radii, so the angles are all the same. This proves this gives a regular polygon.

The lengths of the side will scale with the radius of the circle. For a regular polygon of $n$ sides in a circle of radius $r$, the side will be $s=2r \sin \frac {\pi}n$, which you can prove by looking at the isoceles triangles.

Ross Millikan
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  • @johnny When $r=1$ and $n$ is very large, then $2r \sin \frac{\pi}n$ will be very close to $2\pi/n$, but it will never be exactly equal no matter how big $n$ is. – Erick Wong Jul 27 '16 at 06:10
  • thanks for this awesome answer, could you add a quick schema for the sake of future reader please? – DiaJos Feb 17 '19 at 22:09