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Inoria · Mar 25, 2010, 05:21 AM

Right... I've got a question to find formula for perimeter of regular polygon which has all of its vertices on a circle. If the polygon is n-sides.
I worked out that it will be P=nx [2x (cos(alpha) x c)] where .
I'm not sur if I'm right but then there is a question what I'm struggle with:

Find the number this formula equals as n increases [Hint: Put n= 100, 200, 300,. until you see the pattern]. Put formally I want you to work out lim (P)

ebaines · Mar 25, 2010, 08:57 AM

Hello Inoria. It seeems that part of your post got lost. I assume that $\alpha$ is the half-angle of the "wedge" of a side of the polygon, and that $c$ is the radius of the circle - is that right? If so, then you need to consider the following:

1. The length of one side of the polygon is $2c\cdot sin \alpha.$ Note the use of sin (not cosine).

2. The angle $\alpha$ is related to $n$ by $\alpha = \pi/n$ . Do you see why?

The relationship they're looking for is that as $n$ gets very large the perimeter approaches the perimeter for a circle of radius $c$ .

Inoria · Mar 25, 2010, 09:31 AM

Yes that's correct. Now I can see the relationship. Perfect thank you very much. Just one thing is not clear for me. Why did You use sin. I'm doing this exercise on case of octagon and I have given just an angle and radius.

Could you explain it please.

ebaines · Mar 25, 2010, 10:03 AM

Perhaps the attached pic will help.

Inoria · Mar 25, 2010, 10:16 AM

Oh... all right... thans so much... :)