Circles and polygons

1. Jul 3, 2006

scott_alexsk

This has been bugging me for a while and I thought that you guys might know an answer. Awhile ago I realized that there is a direct relationship between the radius (as in the distance between a corner and the center) squared and the area of any regular polygon with the same number of sides. For example the radius of any square, squared and multiplied by 2 equals the area. But also, for any triangle the radius squared times approx. 1.3 equals the area.

By finding this for several polygons, I found what I expected, the constant for each polygon as the sides increase approaches 3.14. I also found that there is a constant between the diameter of a polygon and the perimeter which also approaches 3.14 as the number of sides increases on the regular polygon.

My question is, according to these relationships is there a way to determine the value of any of these constants for polygons including a circle (3.14) (assuming that it is a polygon with a infinite number of sides) using an equation? Is there any equation you can think of that shows this? Perhaps there is an equation that determines this from the relation of the radius and the apothum (sp)?

Thanks
-scott

Last edited: Jul 3, 2006
2. Jul 3, 2006

MalayInd

If the regular polygon has even no. of sides then
Area= (1/2)*no. of sides* side squared* sin(2pi/n)

Keep Smiling
Malay

3. Jul 3, 2006

0rthodontist

That is also true when the polygon has an odd number of sides.

("side squared" means Scott's "radius" squared)

4. Jul 3, 2006

scott_alexsk

Area= (1/2)*no. of sides* radius squared* sin(2pi/n)

What's n stand for?

Thanks for all of the posts
-scott

5. Jul 4, 2006

MalayInd

n stands for the number of sides.

Keep Smiling
Malay