AMC10 2006 B

AMC10 2006 B · Q6

AMC10 2006 B · Q6. It mainly tests Circle theorems, Area & perimeter.

A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $2/\pi$, as shown. What is the perimeter of this region?

一个区域由在边长为$2/\pi$的正方形边上构造的半圆弧所包围，如图所示。该区域的周长是多少？

(A) $\frac{4}{\pi}$ $\frac{4}{\pi}$

(B) 2 2

(D) 4 4

(E) $\frac{16}{\pi}$ $\frac{16}{\pi}$

Answer

Correct choice: (D)

正确答案：（D）

Solution

Since the square has side length $2/\pi$, the diameter of each circular section is $2/\pi$. The boundary of the region consists of 4 semicircles, whose total perimeter is twice the circumference of a circle having diameter $2/\pi$. Hence the perimeter of the region is $$2 \cdot \left( \pi \cdot \frac{2}{\pi} \right) = 4.$$

由于正方形边长为$2/\pi$，每个圆弧部分的直径为$2/\pi$。该区域的边界由4个半圆组成，其总周长是直径为$2/\pi$的圆周长的两倍。因此，该区域的周长为 $$2 \cdot \left( \pi \cdot \frac{2}{\pi} \right) = 4.$$

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