Question

Andrea 在一个正五边形内切一个圆，并在该五边形外接一个圆，然后计算两个圆之间区域的面积。Bethany 对一个正七边形（ $7$ 条边）做了同样的事情。两个区域的面积分别为 $A$ 和 $B$ ，每个多边形的边长都是 $2$ 。下列哪项正确？

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon ( $7$ sides). The areas of the two regions were $A$ and $B,$ respectively. Each polygon had a side length of $2.$ Which of the following is true?

$A = \dfrac{25}{49}B$

$A = \dfrac{5}{7}B$

$A = B$

$A = \dfrac{7}{5}B$

$A = \dfrac{49}{25}B$

答案：C

知识点：正多边形勾股定理圆面积

难度评级：1910

解答：

对边长为 $2$ 的正多边形，令 $O$ 为中心， $M$ 为一条边的中点， $N$ 为该边的一个端点。那么 $\triangle OMN$ 在 $M$ 处为直角，且 $MN = 1$ ， $OM = r$ （内切圆半径）， $ON = R$ （外接圆半径）。

所以 $R^2 - r^2 = 1$ ，两圆之间的面积为 $\pi(R^2 - r^2) = \pi$ ，与边数无关。因此 $A = B$ 。

因此，正确答案是 C。

For a regular polygon with side length $2,$ let $O$ be the center, $M$ the midpoint of a side, and $N$ an endpoint of that side. Then $\triangle OMN$ has a right angle at $M,$ with $MN = 1,$ $OM = r$ (inradius), and $ON = R$ (circumradius).

So $R^2 - r^2 = 1,$ and the area between the circles is $\pi(R^2 - r^2) = \pi$ for any number of sides. Hence $A = B.$

Thus, the correct answer is C.

2009 AMC 12A 第 19 题

答案：C

难度评级：1910

解答：

其他年份的第 19 题