2006 AMC 10B 第 6 题

Question

如图，一个区域由以边长为

\tfrac{2}{\pi}

的正方形各边为直径构造的半圆弧围成。这个区域的周长是多少？

A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure

\tfrac{2}{\pi},

as shown. What is the perimeter of this region?

\dfrac{4}{\pi}

2

\dfrac{8}{\pi}

4

\dfrac{16}{\pi}

正方形的每条边都是一个半圆弧的直径。

Each side of the square is the diameter of one semicircular arc

直径为

d

的半圆弧长为

\tfrac12\pi d

。

A semicircle on diameter

d

has arc length

\tfrac12\pi d

每条边长为

\tfrac{2}{\pi}

，也是一个半圆弧的直径，所以每条弧长为

\tfrac12\pi\cdot\tfrac{2}{\pi}=1

。

边界由四条这样的弧组成，所以周长为

4\cdot1=4

。

所以正确答案是 D。

Each side has length

\tfrac{2}{\pi},

the diameter of a semicircular arc, so each arc has length

\tfrac12\pi\cdot\tfrac{2}{\pi}=1.

The boundary consists of four such arcs, so the perimeter is

4\cdot1=4.

Thus, the correct answer is D.

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