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

#1**+2 **

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