Experts Exchange > Questions > Calculate the radius of a n-sided polygon such that the area of the polygon matches the area of a circle with the same radius.

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Calculate the radius of a n-sided polygon such that the area of the polygon matches the area of a circle with the same radius.

This creates an n-sided closed polygon whose points lie on a circle of the same radius - you can see this in action at n-sided polygon fits inside circle.

What I would like is actually 2 fold.

1 - To calculate the path of an n-sided polygon so that the circle fits inside the polygon (just like the existing polygon fits inside the circle)
2 - To calculate the path of an n-sided polygon so that the area of the circle and the polygon are the same.

Don't worry about the coding, if you have issues with that, just the maths required will be enough for me.

I'm guessing that all I really need is to adjust the meters/miles conversion as this is all that actually alters the side the polygon.

Regards,

Richard Quadling.

P.S. I'm using Google Chrome V10, so the sliders are not going to be visible in older browsers - simply enter the number and tab off the input seems to work OK for IE/FF. This is just a test script, so not an important issue.

Note: since your code uses degrees, not radians, so am I. If you get really weird results, change all the 360s and 180s to 2pi and pi.
Given a circumradius c (distance from center to vertex) the area

Note: since your code uses degrees, not radians, so am I. If you get really weird results, change all the 360s and 180s to 2pi and pi.

Given a circumradius c (distance from center to vertex) the area is
nc^2sin(360/n)/2
So if the circle's area is A then you want to make your vertices
sqrt(2A/nsin(360/n))
from the center.

To circumscribe the polygon around the circle, you want the apothem a (distance from center to closest point on edge) to be the same as the radius of the circle so you want the vertices to be
r sec(180/n)
from the center (where r is the radius of the circle and sec(180/n)=1/cos(180/n)

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