AMC10 2009 A

AMC10 2009 A · Q6

AMC10 2009 A · Q6. It mainly tests Area & perimeter.

A circle of radius 2 is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?

一个半径为2的圆内切于一个半圆中，如图所示。半圆内圆外阴影部分的面积占半圆面积的几分之几？

(A) $\frac{1}{2}$ $\frac{1}{2}$

(B) $\frac{\pi}{6}$ $\frac{\pi}{6}$

(D) $\frac{2}{5}$ $\frac{2}{5}$

(E) $\frac{3}{\pi}$ $\frac{3}{\pi}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

The semicircle has radius 4 and total area $\frac{1}{2} \cdot \pi \cdot 4^2 = 8\pi$. The area of the circle is $\pi \cdot 2^2 = 4\pi$. The fraction of the area that is not shaded is $\frac{4\pi}{8\pi} = \frac{1}{2}$, and hence the fraction of the area that is shaded is also $\frac{1}{2}$.

半圆半径为4，总面积 $\frac{1}{2} \cdot \pi \cdot 4^2 = 8\pi$。圆的面积为 $\pi \cdot 2^2 = 4\pi$。非阴影面积占比 $\frac{4\pi}{8\pi} = \frac{1}{2}$，因此阴影面积占比也为 $\frac{1}{2}$。

Topics

GE.007 Area & perimeter