2005 AMC 10A 第 12 题

Question

图中形状称为三叶形，由围绕全等等边三角形的边画圆形扇形构成。若三叶形的水平底边长为

2

，它的面积是多少？

The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length

2?

\dfrac{1}{3}\pi + \dfrac{\sqrt{3}}{2}

\dfrac{2}{3}\pi

\dfrac{2}{3}\pi + \dfrac{\sqrt{3}}{4}

\dfrac{2}{3}\pi + \dfrac{\sqrt{3}}{3}

\dfrac{2}{3}\pi + \dfrac{\sqrt{3}}{2}

答案：B

知识点：扇形面积分割等边三角形

难度评级：1460

小提示：

底边长

2

跨过两个半径，所以每个圆形扇形半径为

1

。

The base of length

2

spans two radii, so each circular sector has radius

1

大提示：

将四个等边三角形和四个圆弓形重新组合成四个半径为

1

、圆心角为

60^\circ

的扇形。

Rearrange the four equilateral triangles and four circular segments into four

60^\circ

sectors of radius

1

解答：

底边长

2

等于两个半径，所以半径为

1

。这个三叶形由四个等边三角形和四个圆弓形组成，它们可以重新拼成四个半径为

1

、圆心角为

60^\circ

的扇形，总面积为

4 \cdot \dfrac{60}{360}\pi (1)^2 = \dfrac{2}{3}\pi

。

所以正确答案是 B。

Since the base

2

equals two radii, the radius is

1.

The trefoil is made of four equilateral triangles and four circular segments, which reassemble into four

60^\circ

sectors of a circle of radius

1.

Their total area is

4 \cdot \dfrac{60}{360}\pi (1)^2 = \dfrac{2}{3}\pi.

Thus, the correct answer is B.

2005 AMC 10A 第 12 题

答案：B

难度评级：1460

小提示：

大提示：

解答：

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