A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$? $\textbf{(A) } 1 - \frac{\sqrt{2}}{2} \qquad$ $\textbf{(B) } \frac{\sqrt{2}}{4} \qquad$ $\textbf{(C) } \sqrt{2} - 1 \qquad$ $\textbf{(D) } \frac{1}{2} \qquad$ $\textbf{(E) } \frac{1+\sqrt{2}}{4}$
Solution
Connect $AE, BF$; they are diagonals of the rectangle and diameters of the octagon, meeting at its center $O$.
The diagonals of a rectangle cut it into four triangles of equal area, so $[ABEF]=4[OAB]$; and $[ABCDEFGH]=8[OAB]$.
