Section: 圆圆区域 | 10.10 圆形区域

Section outline

Area of a Circle
::圆圆区域

To find the area of a circle , all you need to know is its radius . If $\begin{align*}r\end{align*}$ is the radius of a circle, then its area is $\begin{align*}A=\pi r^2\end{align*}$ .
::要找到圆的区域, 您只需要知道它的半径。如果 r 是圆的半径, 那么它的区域就是 Ar2 。

We will leave our answers in terms of $\begin{align*}\pi\end{align*}$ , unless otherwise specified.
::除非另有说明,否则我们将以______字来回答问题。

What if you were given the radius or diameter of a circle? How could you find the amount of space the circle takes up?
::如果给您一个圆的半径或直径呢? 您如何找到圆所占用的空间 ?

Examples
::实例

Example 1
::例1

Find the area of a circle with a diameter of 12 cm.
::查找直径为12厘米的圆形区域。

If $\begin{align*}d = 12 \ cm\end{align*}$ , then $\begin{align*}r = 6 \ cm\end{align*}$ . The area is $\begin{align*}A=\pi \left(6^2 \right)=36 \pi \ cm^2\end{align*}$ .
::如果 d= 12 cm, r= 6 cm. 区域为 A ( 62) = 36 cm 2。

Example 2
::例2

If the area of a circle is $\begin{align*}20 \pi\end{align*}$ units, what is the radius?
::如果圆的面积是 20° 单位,圆的半径是多少?

Plug in the area and solve for the radius.
::插入区域并解决半径问题。

\begin{aligned} 20 π & = π r^{2} \\ 20 & = r^{2} \\ r & = \sqrt{20} = 2 \sqrt{5} u n i t s \end{aligned}

$\begin{align*}20 \pi &= \pi r^2\\ 20 &= r^2\\ r &= \sqrt{20}=2 \sqrt{5} units\end{align*}$
::20r220=r2r=20=25单位

Example 3
::例3

A circle is inscribed in a square . Each side of the square is 10 cm long. What is the area of the circle?
::以正方形标出一个圆形。正方形的每侧长10厘米。圆的面积是多少?

The diameter of the circle is the same as the length of a side of the square. Therefore, the radius is 5 cm.
::圆的直径与方形一面的长度相同。因此,半径为5厘米。

\begin{aligned} A = π 5^{2} = 25 π c m^{2} \end{aligned}

$\begin{align*}A=\pi 5^2=25 \pi \ cm^2\end{align*}$
::A52=25平方厘米2

Example 4
::例4

Find the area of the shaded region from Example 3.
::从例3中找到阴影区域的面积。

The area of the shaded region would be the area of the square minus the area of the circle.
::阴影区域的面积为平方减去圆圈区域的面积。

\begin{aligned} A = 10^{2} - 25 π = 100 - 25 π \approx 21.46 c m^{2} \end{aligned}

$\begin{align*}A=10^2-25 \pi =100-25 \pi \approx 21.46 \ cm^2\end{align*}$
::A=102-25100-2521.46 平方厘米2

Example 5
::例5

Find the diameter of a circle with area $\begin{align*}36 \pi\end{align*}$ .
::找到一个圆形直径,区域为36°。

First, use the formula for the area of a circle to solve for the radius of the circle.
::首先,使用圆区域公式来解析圆半径。

\begin{aligned} A & = π r^{2} \\ 36 π & = π r^{2} \\ 36 & = r^{2} \\ r & = 6 \end{aligned}

$\begin{align*}A&=\pi r^2\\ 36 \pi &=\pi r^2\\ 36 &= r^2\\ r&=6\end{align*}$
::r236r236=r2r=6

If the radius is 6 units, then the diameter is 12 units.
::如果半径为6个单位,直径为12个单位。

Review
::回顾

Fill in the following table. Leave all answers in terms of $\begin{align*}\pi\end{align*}$ .
::填写下表,将所有答案保留在_________________________

	*radius*	*Area*	*circumference*
1.	2
2.		$\begin{align}16 \pi\end{align}$
3.			$\begin{align}10\pi\end{align}$
4.			$\begin{align}24\pi\end{align}$
5.	9
6.		$\begin{align}90\pi\end{align}$
7.			$\begin{align}35\pi\end{align}$
8.	$\begin{align}\frac{7}{\pi}\end{align}$
9.			60
10.		36

Find the area of the shaded region. Round your answer to the nearest hundredth.
::找到阴暗区域的区域。将您的答复转至最近的一百个区域。

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

Area of a Circle ::圆圆区域

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)