AMC10 2005 A

AMC10 2005 A · Q12

AMC10 2005 A · Q12. It mainly tests Triangles (properties), Circle theorems.

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?

图中所示图形称为三叶草，由在全等等边三角形的边上绘制圆扇形构成。其水平底边长度为 2 的三叶草的面积是多少？

(A) $\frac{1}{3}\pi + \frac{\sqrt{3}}{2}$ $\frac{1}{3}\pi + \frac{\sqrt{3}}{2}$

(B) $\frac{2}{3}\pi$ $\frac{2}{3}\pi$

(D) $\frac{2}{3}\pi + \frac{\sqrt{3}}{3}$ $\frac{2}{3}\pi + \frac{\sqrt{3}}{3}$

(E) $\frac{2}{3}\pi + \frac{\sqrt{3}}{2}$ $\frac{2}{3}\pi + \frac{\sqrt{3}}{2}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) The trefoil is constructed of four equilateral triangles and four circular segments, as shown. These can be combined to form four $60^\circ$ circular sectors. Since the radius of the circle is $1$, the area of the trefoil is $$ \frac{4}{6}\left(\pi\cdot 1^2\right)=\frac{2}{3}\pi. $$

（B）如图所示，该三叶形由四个等边三角形和四个圆弓形组成。这些可以组合成四个 $60^\circ$ 的扇形。由于圆的半径为 $1$，三叶形的面积为 $$ \frac{4}{6}\left(\pi\cdot 1^2\right)=\frac{2}{3}\pi. $$

Topics