AMC8 2011

AMC8 2011 · Q25

AMC8 2011 · Q25. It mainly tests Area & perimeter, Geometry misc.

A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle’s shaded area to the shaded area between the two squares?

一个半径为1的圆内接于一个正方形，并外接于另一个正方形，如图所示。圆的阴影面积与两个正方形之间阴影面积的比值最接近于哪个分数？

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

(B) 1 1

(D) 2 2

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

Answer

Correct choice: (A)

正确答案：（A）

Solution

The area of the smaller square is one half of the product of its diagonals. Note that the distance from a corner of the smaller square to the center is equivalent to the circle's radius so the diagonal is equal to the diameter: $2 \cdot 2 \cdot \frac{1}{2}=2.$ The circle's shaded area is the area of the smaller square subtracted from the area of the circle: $\pi - 2.$ If you draw the diagonals of the smaller square, you will see that the larger square is split $4$ congruent half-shaded squares. The area between the squares is equal to the area of the smaller square: $2.$ Approximating $\pi$ to $3.14,$ the ratio of the circle's shaded area to the area between the two squares is about \[\frac{\pi-2}{2} \approx \frac{3.14-2}{2} = \frac{1.14}{2} \approx \boxed{\textbf{(A)}\ \frac12}\].

小正方形的面积是其对角线乘积的一半。注意从小正方形角到中心的距离等于圆半径，对角线等于直径：$2 \cdot 2 \cdot \frac{1}{2}=2$。圆的阴影面积是圆面积减小正方形面积：$\pi - 2$。画小正方形对角线，大正方形被分成$4$个全等的半阴影小方形。两个正方形间面积等于小正方形面积：$2$。用$\pi\approx3.14$，比值为 \[\frac{\pi-2}{2} \approx \frac{3.14-2}{2} = \frac{1.14}{2} \approx \boxed{\textbf{(A)}\ \frac12}\]。

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