Natasha carries a paintbrush of length 1 and walks around the perimeter of a regular 2016-gon
with side length 1, painting all of the area outside the polygon that she can reach. What is the
area that Natasha paints?
Choose the correct choice from the following answers:
A) 2014+π
B) 2015+π
C) 2016+π
D) 2013+π
E) 2012+π
If the curve is convex, Natasha will only be turning one way, and by the time she   returns to the origin she will have turned exactly 2π radians. Therefore, she will paint the area   of a circle as she turns corners, because she paints a sector of the unit circle each time until the   sum of the sector angles is 2π. On each of the 2016 straight edges, she paints a rectangle of   length 1 and width 1, because the 2016-gon has side length 1 and the paintbrush also has length   1. Thus the total area is 2016 *1 + π* 12 = 2016 + π   correct answer 2016+π
<result>C</result>
Natasha carries a paintbrush of length 1 and walks around the perimeter of a regular 2016-gon
with side length 1, painting all of the area outside the polygon that she can reach. What is the
area that Natasha paints?