10.8圆的面积

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  圆的面积

  

  [Figure 1]

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  Debra is looking for a table to go in a small breakfast nook in her house. In order to get a table and two chairs in the breakfast nook, the table must be smaller than 2  square  feet in  area . Debra sees a table she likes. The table has a  diameter  of 28 inches. How can Debra figure out if the table is small enough to fit in the breakfast nook?
  ::Debra正在寻找一张桌子,在她家的一个小早餐之屋里。为了在早餐之屋里找到一张桌子和两张椅子,桌子必须小于地区两平方英尺。Debra看到一张她喜欢的桌子。桌子直径28英寸。Debra如何能弄清楚桌子是否小到足以适应早餐之屋?

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  In this concept, you will learn to find areas of circles given the  radius  or the diameter.
  ::在这个概念中,根据半径或直径,您将学会寻找圆圈区域。

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  Finding Area of Circles
  ::查找圆圆区域

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  The inside of a  circle  is called the  area  of the circle. The area is the surface or  space  inside the circle.
  ::圆的内侧称为圆的区域。区域是圆内的表面或空间。

  

  [Figure 2]

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  In this picture, the area of the circle is yellow. As you can see, it is all the space inside the circle.
  ::在此图片中, 圆的面积是黄色的。 正如您所看到的, 它就是圆内的全部空间 。

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  To figure out the area of a circle, you will need a couple of different measurements. The first one is  pi . You will need to use the numerical value for  pi , or 3.14, to represent the  ratio  between the diameter and the  .
  ::要找出圆的面积, 您需要几个不同的测量。 第一个是 pi 。 您需要使用 pi 的数值值, 或者 3. 14 来表示直径和 直径之间的比 。

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  The next measurement you need to use is the  radius , which is the  distance  from the  center  of a circle to the  edge , or half of the circle’s diameter.
  ::您需要使用的下一个测量方法是半径, 也就是圆的中心到边缘的距离, 或者圆的直径的一半。

  

  [Figure 3]

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  To calculate the area of a circle, multiply the radius squared (multiplied by itself) by pi (3.14). Here is the formula.
  ::要计算圆的面积,请将半径平方乘以 pi (3.14) 。 这是公式 。

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

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  Let’s try an example.
  ::让我们以身作则。

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  Find the area of the circle.
  ::查找圆圈的区域 。

  

  [Figure 4]

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  First, substitute the given information into the formula.
  ::首先,将给定信息替换为公式。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& (3.14)(3^2)\end{array}\end{array}$$

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  Next, multiply.
  ::下一个,乘数。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)(9)\\ A& =& 28.26m{m}^2\end{array}\end{array}$$

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  Notice that area is measured in  square units .
  ::注意区域以平方单位计量。

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  If you have been given the diameter and not the radius, you can still figure out the area of the circle.
  ::如果给定了直径而不是半径,您仍然可以找出圆的面积。

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  Start by dividing the diameter in half since the radius is one-half the  measure  of the diameter then use the formula and solve for area.
  ::开始时将直径除以半, 因为半径是直径的半径, 然后使用公式并解决区域 。

  

  [Figure 5]

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  Find the area of the circle with a diameter of 6 inches.
  ::查找圆的直径为6英寸的区域。

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  First, divide the diameter in half to find the radius.
  ::首先,将直径除以一半以找到半径。

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    $\begin{array}{r}6÷2=3\text{inches}=\text{radius}\end{array}$
  ::6 2 = 3 英寸 = 半径

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  Next, substitute the given values into the formula and solve for the area of the circle.
  ::接下来,将给定值替换为公式,并解决圆区域。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& (3.14)(3^2)\\ A& =& (3.14)(9)\\ A& =& 28.26in{.}^2\end{array}\end{array}$$

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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 Debra and the table she wants for her breakfast nook.
  ::早些时候,你得到一个问题 关于黛博拉 和她想要的餐桌 她的早餐 Nook。

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  Debra’s husband measured the space in the breakfast nook and told her the table has to be less than 2 square feet to fit comfortably in the nook. The table Debra likes has a diameter of 28 inches. Debra needs to use the area of a circle formula to find the area of the table.
  ::Debra的丈夫测量了早餐之洞的空间,并告诉她,桌子必须小于2平方英尺才能舒适地适应。 Debra喜欢的桌子直径为28英寸。 Debra需要使用圆形公式的面积来找到桌子的面积。

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  First, she should divide the diameter in half to find the radius.
  ::首先,她应该把直径除以一半 才能找到半径

  $$\begin{array}{r}28÷2=14\text{inches}\end{array}$$

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  The radius is 14 inches.
  ::半径14英寸

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  Next, substitute the given values into the formula.
  ::接下来,将给定值替换为公式。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& (3.14)({14}^2)\end{array}\end{array}$$

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  Next, evaluate the exponent and multiply.
  ::下一步,评价指数和乘法。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)(196)\\ A& =& 615.44{\text{inches}}^2\end{array}\end{array}$$

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  The area of the table is 615.44 square inches.
  ::表格面积为615.44平方英寸。

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  Then, to find out if the table will fit in the nook, convert the 2 square feet into inches.
  ::然后,看看桌子是否合适 将2平方英尺转换成英寸

  $$\begin{array}{rl}& (2\times 12)(2\times 12)\\ & 24\times 24\\ & 576\text{square inches}\end{array}$$

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  The table must be smaller than 576 square inches to fit in the space. The table is too big.
  ::桌子必须小于576平方英寸才能适应空间。桌子太大了。

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  Example 2
  ::例2

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  The diameter of a circle is 13 feet. What is the area of the circle?
  ::圆的直径是13英尺。圆的面积是多少?

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  First, divide the diameter in half to find the radius.
  ::首先,将直径除以一半以找到半径。

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    $\begin{array}{r}13÷2=6.5\text{feet}\end{array}$
  ::132=6.5英尺

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  The radius is 6.5 feet.
  ::半径是6.5英尺

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  Next, substitute the given values into the formula and solve for the area of the circle.
  ::接下来,将给定值替换为公式,并解决圆区域。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& (3.14)({6.5}^2)\\ A& =& (3.14)(42.25)\\ A& =& 132.665ft{.}^2\end{array}\end{array}$$

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  Then, round up the nearest hundredth.
  ::然后把最近的一百个集合起来

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  The answer is 132.67 sq. feet.
  ::答案是132.67平方英尺

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  Example 3
  ::例3

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  Find the area given the radius.
  ::查找给定半径的区域 。

  $$\begin{array}{r}r=12cm\end{array}$$

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  First, substitute the given information into the formula.
  ::首先,将给定信息替换为公式。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& (3.14)({12}^2)\end{array}\end{array}$$

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  Next, evaluate the exponent and multiply.
  ::下一步,评价指数和乘法。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)(144)\\ A& =& 452.16c{m}^2\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}452.16c{m}^2\end{array}$ .
  ::答案是452.16cm 2。

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  Example 4
  ::例4

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  Find the area given the radius.
  ::查找给定半径的区域 。

  $$\begin{array}{r}r=18cm\end{array}$$

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  First, substitute the given information into the formula.
  ::首先,将给定信息替换为公式。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& (3.14)({18}^2)\end{array}\end{array}$$

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  Next, evaluate the exponent and multiply.
  ::下一步,评价指数和乘法。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)(324)\\ A& =& 1017.36c{m}^2\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}1017.36c{m}^2\end{array}$ .
  ::答案是1017.36cm 2。

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  Example 5
  ::例5

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  Find the area given the diameter.
  ::查找给定直径的区域 。

  $$\begin{array}{r}d=14in\end{array}$$

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  First, divide the diameter in half to find the radius.
  ::首先,将直径除以一半以找到半径。

  $$\begin{array}{r}14÷2=7\text{inches}\end{array}$$

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  The radius is 7 inches.
  ::半径是7英寸

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  Next, substitute the given values into the formula.
  ::接下来,将给定值替换为公式。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \pi r^2\\ A& =& (3.14)(7^2)\end{array}\end{array}$$

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  Next, evaluate the exponent and multiply.
  ::下一步,评价指数和乘法。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)(49)\\ A& =& 153.86f{t}^2\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}153.86f{t}^2\end{array}$ .
  
  ::答案是153.86 f t 2。

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  Review
  ::回顾

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  Find the area of the following circles given the radius. Use the value approximated value 3.14 for the number  $\begin{array}{r}\pi .\end{array}$
  ::根据半径查找以下圆圈的面积。使用大约值3.14的数值表示编号 + 。

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  1. $\begin{array}{r}r=4in.\end{array}$  
     ::r = 4 i n 。
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  2. $\begin{array}{r}r=5cm.\end{array}$  
     ::r = 5厘米。
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  3. $\begin{array}{r}r=8in.\end{array}$  
     ::r = 8 i n 。
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  4. $\begin{array}{r}r=2cm.\end{array}$  
     ::r = 2厘米。
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  5. $\begin{array}{r}r=7m.\end{array}$  
     ::r = 7米。
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  6. $\begin{array}{r}r=9in.\end{array}$  
     ::r = 9 i n 。
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  7. $\begin{array}{r}r=10ft.\end{array}$  
     ::r = 10 f t.
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  8. $\begin{array}{r}r=11cm.\end{array}$  
     ::r = 11厘米。
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  9. $\begin{array}{r}r=20ft.\end{array}$  
     ::r = 20英尺吨。
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  10. $\begin{array}{r}r=30mi.\end{array}$  
      ::r = 30米i.

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  Find the area of the following circles given the diameter. Use the value approximated value 3.14 for the number  $\begin{array}{r}\pi .\end{array}$
  ::根据直径查找以下圆圈的面积。使用数值大约为3.14的数值表示 。

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  11. $\begin{array}{r}d=10in.\end{array}$  
      ::d = 10 i n 。
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  12. $\begin{array}{r}d=12m.\end{array}$  
      ::d = 12米。
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  13. $\begin{array}{r}d=14cm.\end{array}$  
      ::d = 14厘米。
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  14. $\begin{array}{r}d=16ft.\end{array}$  
      ::d = 16 f t.
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  15. $\begin{array}{r}d=18in.\end{array}$  
      ::d = 18 i n 。
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  16. $\begin{array}{r}d=22ft.\end{array}$  
      ::d = 22英尺吨。
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  17. $\begin{array}{r}d=24cm.\end{array}$  
      ::d = 24厘米。
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  18. $\begin{array}{r}d=28m.\end{array}$  
      ::d = 28米。
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  19. $\begin{array}{r}d=30m.\end{array}$  
      ::d = 30米。
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  20. $\begin{array}{r}d=36ft.\end{array}$  
      ::d = 36 f t.

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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 。

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  Resources
  ::资源