How to make the general formula for area regular polygon work for circles-any ideas? Watch

jelly fish
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Report Thread starter 12 years ago
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I was wondering whether anyone had any ideas on how to make my general formula for the area ofa regular polygon A = 0.5p{0.5p/[n[tan (180/n)]] work for a circle? I think it might be possible using radians or small angle approximation, but I keep getting stuck on the number of sides, which I can't eliminate frommy equationand seems necessary for polygons. Any thoughts?
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Cexy
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Report 12 years ago
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I can't quite see where you get your expression from - perhaps if you explain then I can help.

The way I would do it would be to imagine dividing up a polygon with n sides into n isosceles triangles, each with two sides of length r and an angle in the middle of 2pi/n. You then use the formula for the area of a triangle:

a = 0.5 ab sin C

where here you have a=b=r, and C=2pi/n (using radians, where 2pi radians = 360 degrees). Therefore the area of each triangle is

a = 0.5r2 sin(2pi/n)

and there are n triangles, so the area of the polygon is

A = na = 0.5 n r2 sin(2pi/n)

To get the area of a circle, you let n increase to infinity. This makes the angle 2pi/n very small, so that you can use the small angle approximation

sin x = x

which gives you

A = 0.5 n r2 2pi/n = pi r2

just as you expect.

Edit: Whoops, didn't see that this was posted twice! Never mind, maybe my post will help someone anyway.
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