AMC12 2003 B

AMC12 2003 B · Q15

AMC12 2003 B · Q15. It mainly tests Area & perimeter, Polygons.

A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$?

正八边形 $ABCDEFGH$ 的面积为 1 平方单位。矩形 $ABEF$ 的面积是多少？

(A) $1 - \frac{\sqrt{2}}{2}$ $1 - \frac{\sqrt{2}}{2}$

(B) $\frac{\sqrt{2}}{4}$ $\frac{\sqrt{2}}{4}$

(D) $\frac{1}{2}$ $\frac{1}{2}$

(E) $1 + \frac{\sqrt{2}}{4}$ $1 + \frac{\sqrt{2}}{4}$

Answer

Correct choice: (D)

正确答案：（D）

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 $\boxed{\textbf{(D)}\ \frac{1}{2}}$.

这里有一种简单的看法，设 $p$ 为周长，$a$ 为边心距：八边形面积：$\frac{ap}{2}=1$。矩形面积：$\frac{p}{8}\times 2a=\dfrac{2ap}{8}=\frac{ap}{4}$。由此可见，八边形的面积是该矩形面积的两倍，因此答案是 $\boxed{\textbf{(D)}\ \frac{1}{2}}$。

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