AMC 8 2009 Problem 9

Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?

$\textbf{(A)}\ 21 \textbf{(B)}\ 23 \textbf{(C)}\ 25 \textbf{(D)}\ 27 \textbf{(E)}\ 29$