Question

一个 $1 \times 2$ 长方形内接于一个半圆，较长的一边在直径上。这个半圆的面积是多少？

A $1 \times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?

$\dfrac{\pi}{2}$

$\dfrac{2\pi}{3}$

$\pi$

$\dfrac{4\pi}{3}$

$\dfrac{5\pi}{3}$

答案：C

知识点：勾股定理圆面积

难度评级：1370

小提示：

半径从直径中点连到长方形的上方顶点

The radius goes from the midpoint of the diameter to a top corner of the rectangle.

大提示：

这条半径是一个 $1$ 乘 $1$ 直角三角形的斜边

That radius is the hypotenuse of a $1$ by $1$ right triangle.

视频讲解：

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文字解答：

由勾股定理，半圆半径为 $\sqrt{2}$ 。因此半圆面积为 $\dfrac{1}{2} \cdot \pi \cdot \sqrt{2}^2 = \pi\text{。}$ 所以正确答案是 C。

Using the Pythagorean theorem, we get that the radius of the semicircle is $\sqrt{2}.$ This means that the area of the semicircle is $\dfrac{1}{2} \cdot \pi \cdot \sqrt{2}^2 = \pi.$ Thus, C is the correct answer.

2013 AMC 8 第 20 题