2007 AMC 10A 第 15 题

Question

如图，四个半径为

1

的圆各自与正方形的两条边相切，并且都与一个半径为

2

的圆外切。正方形的面积是多少？

Four circles of radius

1

are each tangent to two sides of a square and externally tangent to a circle of radius

2,

as shown. What is the area of the square?

32

22 + 12\sqrt{2}

16 + 16\sqrt{3}

48

36 + 16\sqrt{2}

连接大圆圆心与两个相邻小圆的圆心。

Connect the center of the large circle to the centers of two adjacent small circles

这个三角形是等腰直角三角形，直角边为

2 + 1 = 3

，所以斜边为

3\sqrt2

。

That triangle is isosceles right with legs

2 + 1 = 3,

so its hypotenuse is

3\sqrt2

考虑连接半径

2

的圆心和两个相邻小圆圆心形成的等腰直角三角形。它的两条直角边长为

2 + 1 = 3

，所以斜边为

3\sqrt2

。

正方形边长比这条斜边多

2

（两端各一个小圆半径），所以

s = 2 + 3\sqrt2

。

面积为

\begin{aligned} (2 + 3\sqrt2)^2 &= 4 + 12\sqrt2 + 18 \\ &= 22 + 12\sqrt2 \end{aligned}\text{。}

所以正确答案是 B。

Consider the isosceles right triangle joining the center of the radius-

2

circle to the centers of two adjacent small circles. Its legs have length

2 + 1 = 3,

so its hypotenuse is

3\sqrt2.

The side of the square exceeds this hypotenuse by

2

(one radius on each end), so

s = 2 + 3\sqrt2.

The area is

\begin{aligned} (2 + 3\sqrt2)^2 &= 4 + 12\sqrt2 + 18 \\ &= 22 + 12\sqrt2. \end{aligned}

Thus, the correct answer is B.

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