AMC10 2003 10B, Problem 23

A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$? [asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);[/asy]

Solution: Here is an easy way to look at this, where $p$ is the perimeter, and $a$ is the apothem: Area of Octagon: $\frac{ap}{2}=1$. Area of Rectangle: $\frac{p}{8}\times 2a=\dfrac{2ap}{8}=\frac{ap}{4}$. You can see from this that the octagon's area is twice as large as the rectangle's area is $\fbox{\frac{1}{2}}$.

Answer: \frac{1}{2}