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# Funky Circle Areas

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Let \(S\) be the union of the set of all points inside a regular nonagon with side length \(2\) units and the set of all points less than \(1\) unit away from a point on the perimeter of the nonagon. What, in units, is the perimeter of \(S\)?

benjamingu22  Nov 10, 2017
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Here is a pic :

The circle has a radius of 1

B is a vertex of the nonagon......and.....  AE, BD , FB, GC, DB, FB  = 1

AB, BC  = 2 =  two edges of the nonagon

And DBF  = 40°

And ED + DF =  (1/9)"S"

"S"  will consist of  a perimeter of  nine "straight" sides, each with a length of 2, as well as  nine   40° arcs that will total to the perimeter of a circle with a radius of 1, i.e., 2pi

So.....the total perimeter is  9 * 2  + 2 * pi  =  2 [ 9 + pi ]  ≈ 46.27 units

CPhill  Nov 11, 2017
edited by CPhill  Nov 11, 2017

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