2009 AMC 12A 第 19 题

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

小提示：

对边长为

2

的正多边形，设内切圆半径和外接圆半径分别为

r

和

R

，并观察中心、一条边的中点和该边的端点。

For a regular polygon with side

2,

let

r

and

R

be the inradius and circumradius, and look at the center, a side’s midpoint, and its endpoint

大提示：

这个直角三角形的两条直角边为

r

和

1

，斜边为

R

，所以无论边数是多少，

R^2 - r^2 = 1

。

That right triangle has legs

r

and

1

and hypotenuse

R,

so

R^2 - r^2 = 1

regardless of the number of sides

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

小提示：

大提示：

解答：

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