HELP ON SECTORS

There are seven rectangular sections, each of area 1, and seven curved sections each of which form one seventh of a sector of a unit circle.  Hence the area of S is 7 + 7*pi*1²/7 or S = 7 + pi

.

We are given a regular heptagon of side length 1. Let S be the set of points that are within a distance of 1 from some point on or inside the heptagon, but not including the heptagon itself. Find the area of S

$\begin{array}{|rcll|} \hline A_1 &=& 1\cdot 1 \\ &=& 1 \\\\ A_2 &=& \pi \cdot 1^2 \cdot \frac{ \frac{360^{\circ}}{7} } {360^{\circ}} \\ &=& \pi \cdot \frac{1}{7} \\\\ S &=& 7\cdot A_1 + 7 \cdot A_2 \\ &=& 7\cdot 1 + 7 \cdot \pi \cdot \frac{1}{7} \\ &=& 7+\pi \\ &=& 10.1415926536 \\ \hline \end{array}$