1958 AMC 12 第 16 题

Question

正六边形内切圆的面积为

100\pi

。该正六边形的面积为：

The area of a circle inscribed in a regular hexagon is

100\pi.

The area of the hexagon is:

600

300

200\sqrt2

200\sqrt3

120\sqrt5

圆的半径就是正六边形的边心距

The circle’s radius is the apothem of the regular hexagon

当边心距为

r

时，六边形边长为

2r\tan30^\circ

For apothem

r,

the hexagon side is

2r\tan30^\circ

圆的半径为

r=10

，也是六边形的边心距。因此边长为

2r\tan30^\circ=\frac{20}{\sqrt3}\text{。}

利用面积等于周长与边心距乘积的一半，六边形的面积为

\frac12\left(6\cdot\frac{20}{\sqrt3}\right)(10) =200\sqrt3\text{。}

所以正确答案为 D。

The circle has radius

r=10,

which is the hexagon’s apothem. Thus the side length is

2r\tan30^\circ=\frac{20}{\sqrt3}.

Using one-half the product of perimeter and apothem, the hexagon’s area is

\frac12\left(6\cdot\frac{20}{\sqrt3}\right)(10) =200\sqrt3.

Therefore, the correct answer is D.

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