Question

一个直径为 $1$ 的半圆位于一个直径为 $2$ 的半圆顶部，如图所示。位于较小半圆内部且位于较大半圆外部的阴影区域称为弓月形。求这个弓月形的面积。

A semicircle of diameter $1$ sits at the top of a semicircle of diameter $2,$ as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.

$\dfrac{1}{6}\pi - \dfrac{\sqrt{3}}{4}$

$\dfrac{\sqrt{3}}{4} - \dfrac{1}{12}\pi$

$\dfrac{\sqrt{3}}{4} - \dfrac{1}{24}\pi$

$\dfrac{\sqrt{3}}{4} + \dfrac{1}{24}\pi$

$\dfrac{\sqrt{3}}{4} + \dfrac{1}{12}\pi$

答案：C

知识点：扇形面积分割等边三角形

难度评级：1660

解答：

小半圆的直径是大圆中的一条长为 $1$ 的弦。将这条弦的两端与大圆圆心相连，得到边长为 $1$ 的等边三角形，面积为 $\dfrac{\sqrt{3}}{4}$ 。

弦与小弧之间的区域连同这个三角形，面积为 $\dfrac{\sqrt{3}}{4} + \dfrac{1}{2}\pi\left(\dfrac{1}{2}\right)^2 = \dfrac{\sqrt{3}}{4} + \dfrac{\pi}{8}$ 。

减去大圆中 $60^\circ$ 扇形面积 $\dfrac{1}{6}\pi(1)^2 = \dfrac{\pi}{6}$ ，得到弓月形面积为 $\dfrac{\sqrt{3}}{4} + \dfrac{\pi}{8} - \dfrac{\pi}{6} = \dfrac{\sqrt{3}}{4} - \dfrac{\pi}{24}$ 。

所以正确答案是 C。

The small semicircle's diameter is a chord of length $1$ in the large circle. Joining its endpoints to the large circle's center gives an equilateral triangle of side $1$ and area $\dfrac{\sqrt{3}}{4}.$

The region between the chord and the small arc, taken together with that triangle, has area $\dfrac{\sqrt{3}}{4} + \dfrac{1}{2}\pi\left(\dfrac{1}{2}\right)^2 = \dfrac{\sqrt{3}}{4} + \dfrac{\pi}{8}.$

Subtracting the $60^\circ$ sector of the large circle, of area $\dfrac{1}{6}\pi(1)^2 = \dfrac{\pi}{6},$ leaves the lune: $\dfrac{\sqrt{3}}{4} + \dfrac{\pi}{8} - \dfrac{\pi}{6} = \dfrac{\sqrt{3}}{4} - \dfrac{\pi}{24}.$

Thus, the correct answer is C.

2003 AMC 10A 第 19 题

答案：C

难度评级：1660

解答：

其他年份的第 19 题