1956 AMC 12 第 35 题

Question

一个菱形由某圆的两条半径和两条弦组成，该圆的半径为

16

英尺。这个菱形的面积为多少平方英尺？

A rhombus is formed by two radii and two chords of a circle whose radius is

16

feet. The area of the rhombus in square feet is:

128

128\sqrt3

256

512

512\sqrt3

菱形的每条边长都是

16

，所以每条弦都等于半径

Every side of the rhombus has length

16,

so each chord equals the radius

长度等于半径的弦所对的圆心角为

60^\circ

；使用公式

s^2\sin\theta

A chord equal to the radius subtends a

60^\circ

central angle; use

s^2\sin\theta

菱形的四条边都等于圆的半径

16

。一条长度等于半径的弦与连接其两个端点的两条半径组成等边三角形，所以所夹的圆心角为

60^\circ

。因此，菱形的面积为

\begin{aligned} 16^2\sin60^\circ &=256\cdot\frac{\sqrt3}{2}\\ &=128\sqrt3 \end{aligned}\text{。}

因此，正确答案是 B。

All four sides of the rhombus equal the circle’s radius,

16.

A chord of length equal to the radius forms an equilateral triangle with the two radii to its endpoints, so the included central angle is

60^\circ.

Therefore the rhombus has area

\begin{aligned} 16^2\sin60^\circ &=256\cdot\frac{\sqrt3}{2}\\ &=128\sqrt3. \end{aligned}

Thus, the correct answer is B.

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