AMC 8 1986 Problem 23
AMC8 1986 年第 23 题
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The large circle has diameter . The two small circles have their centers on and just touch at , the center of the large circle. If each small circle has radius 1 , what is the value of the ratio of the area of the shaded region to the area of one of the small circles?
大圆的直径为 。两个小圆的圆心在 上，且恰好在 点（大圆的圆心）相切。如果每个小圆的半径为 1，着色区域与一个小圆的面积之比是多少？
- A. / 在 和 1 之间
- B.1
- C. / 在 1 和 之间
- D. / 在 和 2 之间
- E. / 无法由所给信息确定
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
The small circle has radius 1, thus its area is .
The large circle has radius 2, thus its area is .
The area of the semicircle above is then .
The part that is not shaded are two small semicircles. Together, these form one small circle, hence their total area is . This means that the area of the shaded part is . This is equal to the area of a small circle, hence the correct answer is 1.
Let be the area of one of the small circles. Since all circles are similar, the large circle's area is . So, the area inside the large circle but outside the small circles is 4A - 2A = 2A. Therefore, the shaded region must then be , so the desired ratio is .
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