AMC 8 2000 Problem 19

AMC8 2000 年第 19 题

圆★★★☆☆

Three circular arcs of radius 5 units bound the region shown. Arcs

AB

and

AD

are quarter-circles, and arc

BCD

is a semicircle. What is the area, in square units, of the region?

三段半径为 5 个单位长的圆弧围成图示区域。弧 AB 和 AD 均为四分之一圆，弧 BCD 是个半圆。那么这个区域的面积是多少平方单位？

正确答案 · Correct Answer

解析 · Solution

Solution 1

Draw two squares: one that has opposing corners at

A

and

B

, and one that has opposing corners at

A

and

D

. These squares share side

\overline{AO}

, where

O

is the center of the large semicircle. These two squares have a total area of

2 \cdot 5^2

, but have two quarter circle "bites" of radius 5 that must be removed. Thus, the bottom part of the figure has area

\displaystyle 2\cdot 25 - 2 \cdot \frac{1}{4}\pi \cdot 5^2 = 50 - \frac{25\pi}{2}

. This is the area of the part of the figure underneath

\overline{BD}

. The part of the figure over

\overline{BD}

is just a semicircle with radius 5, which has area

\displaystyle \frac{1}{2}\pi\cdot 5^2 = \frac{25\pi}{2}

. Adding the two areas gives a total area of

\textbf{(C)}50

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