8.4 圆圈面积

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  圆圆区域

  

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  You have a recipe for an 8-in pizza but you want to make a pizza for your 16-in round pan. Will doubling your recipe be enough to fill the larger pan?
  ::你有一个食谱 一个8英寸的比萨饼 但你想做一个比萨饼 为你的16英寸圆锅。

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  In this concept, you will learn to calculate the area of circles and the area of sectors of circles.
  ::在这个概念中,你将学会计算圆圈面积和圆圈区。

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  Area
  ::地区面积

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  Area is the amount of two-dimensional space a figure takes up. In other words, area is the space contained within a circle’s . The formula for the area of a circle is $\begin{array}{r}A=\pi r^2\end{array}$ .
  ::区域是一个数字占用的二维空间的大小。换句话说,区域是一个圆圈内的空间。圆圈区域的公式是 Ar2。

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  A sector is a part of a circle with radii for two sides and part of the curved circumference as another. Sectors look like pie slices. Look at the diagram below. The shaded area is the sector of the circle.
  ::一个扇区是一个圆圈的一部分,有两面的半径,也有两侧的曲线环绕的一部分。各区看起来像饼切片。看看下面的图表。阴暗区域是圆的扇区。

  

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  For now, the area of a sector can be found by finding the proportion of the circle. In the diagram above, the blue sector is $\begin{array}{r}\frac{1}{4}\end{array}$  of the entire circle.
  ::目前,通过找到圆的比重,可以找到某一区段的面积。在上图中,蓝色区段是整个圆圈的14个。

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  Let’s look at an example.
  ::让我们举个例子。

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  What is the area of the circle below?
  ::下面圆的面积是多少?

  

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  First, substitute what you know into the formula for the area of a circle.
  ::首先,将您所知道的替换为公式, 以取代圆区域 。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& \pi (12{)}^2\end{array}\end{array}$$

  
  ::Ar2A(12)2

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  Next, solve for the area.
  ::下一个,解决这个区域。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi (12{)}^2\\ A& =& \pi (144)\\ A& =& 452.4\end{array}\end{array}$$

  
  :12)2A(144)A=452.4

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  The answer is 452.4.
  ::答案是452.4。

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  The area of the circle is 452.4 cm ² .
  ::圆的面积是452.4厘米2。

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  Let’s look at another example.
  ::让我们再看看另一个例子。

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  The area of a circle is 113.04 square inches. What is its radius ?
  ::圆的面积是113.04平方英寸。圆的半径是多少?

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  First, substitute what you know into the formula for the area of a circle.
  ::首先,将您所知道的替换为公式, 以取代圆区域 。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ 113.04& =& \pi r^2\end{array}\end{array}$$

  
  ::2113.04r2

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  Next, divide both sides by $\begin{array}{r}\pi \end{array}$ .
  ::接下来,将双方除以 。

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  $$\begin{array}{r}\begin{array}{rcl}113.04& =& \pi r^2\\ \frac{113.04}{\pi }& =& \frac{\pi r^2}{\pi }\\ r^2& =& 35.98\end{array}\end{array}$$

  
  ::113.042113.04r2r2=35.98

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  Then, take the square root of both sides to solve for $\begin{array}{r}r\end{array}$ .
  ::然后,以双方的平方根 解决r。

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  $$\begin{array}{r}\begin{array}{rcl}{r}^2& =& 35.98\\ r& =& \sqrt{35.98}\\ r& =& 6\end{array}\end{array}$$

  
  ::r2=35.98r=35.98r=6

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  The answer is 6.
  ::答案是6个

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  The radius of the circle is 6 in.
  ::圆的半径是 6 英寸 。

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  Let’s look at an example of finding the area of a sector.
  ::让我们看看一个找寻一个部门领域的例子。

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  What is the area of the figure below?
  ::下图的方面是什么?

  

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  First, substitute what you know into the formula for the area of a circle.
  ::首先,将您所知道的替换为公式, 以取代圆区域 。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& \pi (8.5{)}^2\end{array}\end{array}$$

  
  ::Ar2A(8.5)2

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  Next, solve for the area.
  ::下一个,解决这个区域。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi (8.5{)}^2\\ A& =& \pi (72.25)\\ A& =& 226.98\end{array}\end{array}$$

  
  :8.5)2A(72.25)A=226.98

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  Then, knowing the figure is one-quarter of a circle, divide this area by 4.
  ::然后,知道这个数字是圆的四分之一, 将这个区域除以4。

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  $$\begin{array}{r}\begin{array}{rcl}{A}_{\text{sector}}& =& \frac{226.98}{4}\\ A_{\text{sector}}& =& 56.7\end{array}\end{array}$$

  
  ::A部门=226.984A部门=56.7

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  The answer is 56.7.
  ::答案是56.7

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  The area of the sector is 56.7 in ² .
  ::该部门面积为56.7英寸2。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about the expanding pizza.
  ::早些时候,有人给了你一个问题 关于扩大比萨饼。

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  You want to know if you simply need to double the ingredients for your pizza if you want to use a 16-in round pan rather than an 8-in round pan.
  ::你想知道,如果你想用16英寸的圆盘而不是8英寸的圆盘,你是否只需要将比萨饼的配料翻一倍就可以了。

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  First, find the radius of each pizza pan.
  ::首先,找到每个披萨盘的半径

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  $$\begin{array}{r}\begin{array}{rcl}& & 8-\text{in round pan}16-\text{in round pan}\\ & & r=\frac{d}{2}r=\frac{d}{2}\\ & & r=\frac{8}{2}r=\frac{16}{2}\\ & & r=4r=8\end{array}\end{array}$$

  
  ::8 - 圆泛泛泛16- 圆泛泛色 = d2 r= d2r= 82 r= 162r= 4 r= 8

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  Next, find the area of each pizza pan.
  ::接下来,找到每个披萨盘的面积

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  $$\begin{array}{r}\begin{array}{rcl}& & 8-\text{in round pan}16-\text{in round pan}\\ & & A=\pi r^{2}A=\pi r^2\\ & & A=\pi (4{)}^{2}A=\pi (8{)}^2\\ & & A=\pi \times 16A=\pi \times 64\\ & & A=50.3A=201.1\end{array}\end{array}$$

  
  ::8 - 整回合Pan16- 整回合 panA2 Ar2A(4)2 A(8)2A16A64A=50.3A=201.1

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  Then, divide the area of the 16-in round pan by the area of the 8-in round pan.
  ::然后,将16英寸圆板的面积除以8英寸圆板的面积。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{A_{16-\text{in round pan}}}{A_{8-\text{in round pan}}}& =& \frac{201.1}{50.3}\\ \frac{A_{16-\text{in round pan}}}{A_{8-\text{in round pan}}}& =& 4\end{array}\end{array}$$

  
  ::A16-in 圆泛A8-in 圆泛泛A8=201.150.3A16-in 圆泛A8-in 圆泛泛泛A8=4

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  The answer is 4.
  ::答案是4。

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  The 16-in pizza pan is four times the size of the 8-in pizza pan, so doubling the ingredients is not enough.
  ::16英寸比萨饼盘是8英寸比萨饼盘的四倍, 所以把成分翻倍是不够的。

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  Example 2
  ::例2

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  What is the area of a circle with a diameter of 45 centimeters?
  ::直径45厘米的圆圈面积是多少?

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  First, find the radius of the circle. Remember the radius is half the diameter.
  ::首先,找到圆的半径。记住半径是直径的一半。

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  $$\begin{array}{r}\begin{array}{rcl}r& =& \frac{d}{2}\\ r& =& \frac{45}{2}\\ r& =& 22.5\end{array}\end{array}$$

  
  ::= rd2r= 452r=22.5

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  Next, substitute what you know into the formula for the area of a circle.
  ::接下来,将您所知道的替换为圆形区域的公式。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& \pi (22.5{)}^2\end{array}\end{array}$$

  
  ::Ar2A(22.5)2

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  Then, solve for the area.
  ::然后,解决这个区域。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi (22.5{)}^2\\ A& =& \pi (506.25)\\ A& =& 1590.4\end{array}\end{array}$$

  
  :22.5)2A(506.25)A=1590.4

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  The answer is 1590.4.
  ::答案是1590.4。

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  The area of the circle is 1590 cm ² .
  ::圆区域为 1590 cm2。

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  Find the area of each circle by using the given dimension.
  ::使用给定的维度查找每个圆的面积。

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  Example 3
  ::例3

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  Find the area of each circle with a radius of 2.5 inches.
  ::查找每个圆的半径为2.5英寸的区域。

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  First, substitute what you know into the formula for the area of a circle.
  ::首先,将您所知道的替换为公式, 以取代圆区域 。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& \pi (2.5{)}^2\end{array}\end{array}$$

  
  ::Ar2A(2.5)2

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  Next, solve for the area.
  ::下一个,解决这个区域。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi (2.5{)}^2\\ A& =& \pi (6.25)\\ A& =& 19.6\end{array}\end{array}$$

  
  ::A(2.5)2A(6.25)A=19.6

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  The answer is 19.6. 
  ::答案是19.6。

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  The area of the circle is 19.6 in ² .
  ::圆区域为19.6英寸2。

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  Example 4
  ::例4

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  Find the area of each circle with a radius of 3 feet.
  ::查找每个圆圈的面积,半径为3英尺。

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  First, substitute what you know into the formula for the area of a circle.
  ::首先,将您所知道的替换为公式, 以取代圆区域 。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& \pi (3{)}^2\end{array}\end{array}$$

  
  ::2A(3)2

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  Next, solve for the area.
  ::下一个,解决这个区域。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi (3{)}^2\\ A& =& \pi (9)\\ A& =& 28.3\end{array}\end{array}$$

  
  ::A(3)2A(9)A=28.3

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  The answer is 28.3.
  ::答案是28.3

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  The area of the circle is 28.3 ft ² .
  ::圆的面积为28.3平方英尺。

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  Example 5
  ::例5

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  Find the area of each circle with a diameter of 8 inches.
  ::查找每个圆的直径为8英寸的区域。

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  First, find the radius of the circle. Remember, the radius is half the diameter.
  ::首先,找到圆的半径。记住,半径是直径的一半。

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  $$\begin{array}{r}\begin{array}{rcl}r& =& \frac{d}{2}\\ r& =& \frac{8}{2}\\ r& =& 4\end{array}\end{array}$$

  
  ::rd2r=82r=4

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  Next, substitute what you know into the formula for the area of a circle.
  ::接下来,将您所知道的替换为圆形区域的公式。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& \pi (4{)}^2\end{array}\end{array}$$

  
  ::2A(4)2

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  Then, solve for the area.
  ::然后,解决这个区域。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi (4{)}^2\\ A& =& \pi (16)\\ A& =& 50.3\end{array}\end{array}$$

  
  ::A(4)2A(16)A=50.3

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  The answer is 50.3.
  ::答案是50.3

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  The area of the circle is 50.3 in ² .
  ::圆的面积是50.3英寸2。

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  Review
  ::回顾

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  Find the area of each circle given the radius. Use 3.14 to approximate $\begin{array}{r}\pi \end{array}$  and round to the nearest hundredths place if needed.
  ::查找给定半径的每个圆的面积。 如果需要, 请使用 3. 14 到 ° 和 圆 到 最近的百 位 。

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  1.  $\begin{array}{r}r=4in\end{array}$
  ::1. r=4 英寸

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  2.  $\begin{array}{r}r=3ft\end{array}$
  ::2. r=3英尺

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  3.  $\begin{array}{r}r=2.5in\end{array}$
  ::3. r=2.5英寸

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  4.  $\begin{array}{r}r=5cm\end{array}$
  ::4. r=5厘米

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  5.  $\begin{array}{r}r=3.5in\end{array}$
  ::5. r=3.5英寸

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  6.  $\begin{array}{r}r=9mm\end{array}$
  ::6. r=9毫米

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  7.  $\begin{array}{r}r=11cm\end{array}$
  ::7 r= 11厘米

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  8.  $\begin{array}{r}r=10in\end{array}$
  ::8 r= 10 英寸

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  9.  $\begin{array}{r}r=7ft\end{array}$
  ::9. r=7吨

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  10.  $\begin{array}{r}r=8in\end{array}$
  ::10 r= 8 英寸

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  Find the area of each sector given the radius and the angle measure. Use 3.14 to approximate $\begin{array}{r}\pi \end{array}$  and round to the nearest hundredths place if needed.
  ::查找每个区段的面积, 给其半径和角度量值。 如果需要, 使用3. 14 到 ° 左右, 圆到最近的一百个位置 。

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  11. 45° angle with a radius of 3 in.
  ::11. 45度角,半径为3英寸。

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  12. 55° angle with a radius of 4 mm
  ::12. 55度角,半径4毫米

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  13. 60° angle with a radius of 5 cm
  ::13. 60度角,半径5厘米

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  14. 43° angle with a radius of 6 in
  ::14.43度角,半径为6英寸

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  15. 70° angle with a radius of 2 in.
  ::15. 70度角,半径为2英寸。

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  Review (Answers)
  ::回顾(答复)

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  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 。