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

1. 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 from my equation and seems necessary for polygons. Any thoughts?
2. I'm guessing p is the side length in your equation, so you have the formula (after converting to radians). It's easy to use a bit of trig to show that this is the same as , where r is the radius of the inscribed circle (i.e. the distance from the centre to the middle of a side). Then as n goes to infinity, , so the formula drops out.
3. Actually p was my perimeter - will your formula still work? Also, does that formula work for polygons as well as circles? THANKS SO MUCH!

Let s be the length of one side. Then p = ns, and so your equation becomes . Then trigonometry on the right-angled triangle with angle , opposite side s/2 and adjacent side r (where r is as in my last post; draw a diagram of a regular polygon to see where this triangle comes from - the vertex of the angle specified is at the centre of the circle) enables you to transform the equation to what I gave previously.

What do you mean by working for polygons? That formula is just a different way of expressing your original formula, in terms of a different variable r, the inscribed circle radius, rather than the perimeter p.
5. This is what i did for my piece of coursework. The general formula for a regular polygon's area dictates that the fact that you increase the number of sides you hit a point where the area does not increase anymore. If you have every seen an animation of a polygon where the number of sides increases you'd see the more and more sides you increase the more it becomes a circle. So as you reach infinity sides you get a perfect circle. But if you did say 1000000 sides you'd probably get the same answer. As n (no of sides) tends to infinity it becomes a circle.

Its the question of whether a circle is just a infinitely sided shape.
6. Thanks guys - and can anyone give me some more specific help on 'the fencing problem', I just really need some algebra to put in it etc
7. If you're referring to this thread, then I'm not sure what else there is to say - the algebra's all been laid out there. Is there a specific part of it you're having a problem with?
8. I was taught ages back that the "general area" of a regular polygon is given by the formula:

A =

where r is the radius of the polygon and n is the number of sides. This got me thinking and if you compare it with the area of a circle, A = then you have:

0.5nSin(360/n) = pi as n tends to infinity (a circle has infinite number of sides). So I was pretty excited as I thought I'd hit on an approximation for pi or something... until I did further math last year and realized if you just use the Maclaurin series for Sin, it all cancels down to pi. I dno if this is even relevant or helpful, but something I thought I'd share.
9. (Original post by Luce the maths boff! not!)
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 from my equation and seems necessary for polygons. Any thoughts?
Your formula is Ok. Here is an excel worksheet that works out the area when number of sides and perimeter are inserted.

I have saved the file as csv . I hope it works.

EDIT. I don't think the csv excel document opens correctly. I have inserted the excel sheet into a word document. If you click then i hope an excel sheet will open. If you then type in the yellow cells you will get the area. i have used YOUR formula with the extra bit to change degrees into radians.
Attached Files
10. Area.csv (350 Bytes, 55 views)
11. Area WORD excel.doc (26.5 KB, 84 views)
12. thanks so so so much guys - that spreadsheet was incredibly useful! im ment to hand the c/w in tommorrow andi've done the 'bare bones' and found the formula,but apparently to get higher marks we have to do lots of algebra and stuff to prove everything we say not just for the formula, and wehave to go beyond GCSE stuff. Can anyone who's done this give me a hand? THANK YOU SO SO SO MUCH!!!
13. I did this coursework and some of the stuff mentioned on this thread is way beyond anything I looked at.

And I hate to be a bore but some of this borders on too much help for coursework

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