2.6 一个区段的面积

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  部门面积

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  While eating lunch with your friends one day you decide to get some pie. The school cafeteria has a sale on pies if you buy the whole pie instead of an individual slice. Since you and your friends plan to eat the whole pie anyway, you are happy to make the purchase.
  ::在学校食堂里,如果购买整份派,而不是个别切片,就会有派的销售。既然你和你的朋友打算吃整份派,你很乐意买。

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  You decide to cut yourself a piece. When you have taken your section of pie out, one of your friends objects that you have cut out more than your fair share. If the radius of the pie is six inches, and the angle formed by the end of your pie wedge is $\begin{array}{r}{30}^{\circ }\end{array}$ , what is the area of your pie piece? Is it more than your fair share if there are five people (including you) splitting the pie?
  ::你决定自己切一块。当你把你的部分派拿出来的时候,你的一个朋友会反对你削减的比公平份额更多的。如果馅饼的半径是6英寸,而你的馅饼的尾端所形成的角度是30英寸,那么你的馅饼的面积是多少?如果有五个人(包括你)分割馅饼,那么它是不是比你的公平份额还多?

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  Area of a Sector
  ::部门面积

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  One of the most common geometric formulas is the area of a circle:
  ::最常见的几何公式之一是圆的面积:

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

  
  ::阿 r2

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  In terms of angle rotation, this is the area created by $\begin{array}{r}2\pi \end{array}$ radians.
  ::就角旋转而言,这是以 2弧度创建的区域。

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  $$\begin{array}{r}2\pi \text{rad}=\pi r^2\text{area}\end{array}$$

  
  ::2 radr2 区域

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  A half-circle, or $\begin{array}{r}\pi \end{array}$ radian rotation would create a section, or sector of the circle equal to half the area or:
  ::半圆圈,或弧度旋转,将产生一个区域或圆圈的区块,等于区域的一半,或:

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  $$\begin{array}{r}\frac{1}{2}\pi r^2\end{array}$$

  
  ::12°r2

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  So an angle of 1 radian would define an area of a sector equal to:
  ::1 弧度角将定义一个部门区域, 其面积等于 :

  

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  $\begin{array}{r}1=\frac{1}{2}{r}^2\end{array}$
  ::1=12r2

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  From this we can determine the area of the sector created by any angle, $\begin{array}{r}\theta \end{array}$ radians, to be:
  ::从这一点上,我们可以确定由任何角度创造的部门面积,即弧度,即:

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  $$\begin{array}{r}A=\frac{1}{2}{r}^2\theta \end{array}$$

  
  ::A=12r2

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  Finding the Area
  ::寻找“区域”

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  1. Crops are often grown using a technique called center pivot irrigation that results in circular shaped fields.
  ::1. 作物的种植往往采用称为中央支流灌溉的技术,这种技术可形成圆形田地。

  

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  Here is a satellite image taken over fields in Kansas that use this type of irrigation system.
  ::这是在堪萨斯州 使用这种灌溉系统的 卫星图象

  

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  If the irrigation pipe is 450 m in length, what is the area that can be irrigated after a rotation of $\begin{array}{r}\frac{2\pi }{3}\end{array}$ radians?
  ::如果灌溉管道长450米,那么在2+3弧度旋转后可以灌溉的面积是多少?

  

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  Using the formula:
  ::使用公式 :

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}(450{)}^2\left(\frac{2\pi }{3}\right)\end{array}$$

  
  ::A=12r2A=12(4502(23)

  

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  The area is approximately 212,058 square meters.
  ::该地区面积约为212,058平方米。

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  2. A doughnut has a hole in the middle with a radius of 1 cm, and the distance from the center of the hole to the outer edge of the doughnut is 3 cm. What is the area of a sector of $\begin{array}{r}\frac{1}{4}\end{array}$ of the doughnut?
  ::2. 甜甜圈中间有一个圆洞,半径为1厘米,从洞中心到甜甜圈外部边缘的距离是3厘米。 甜甜圈中14个区段的面积是多少?

  

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  The formula for the area of a sector is

  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \end{array}$$

  . Using this formula to find the area of the sector from the center outward gives:
  ::扇区面积的公式为 A= 12r2。 使用此公式从中间向外查找扇区区域时, 给出 :

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}{3}^2\frac{\pi }{2}\\ A& =\frac{9\pi }{4}\end{array}$$

  
  ::A=12r2A=12322A=94

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  Now it is necessary to subtract the area of the sector that is part of the hole, and therefore not part of the doughnut:
  ::现在必须减去作为洞的一部分的部门面积, 因此不是甜甜圈的一部分:

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}(1{)}^2\frac{\pi }{2}\\ A& =\frac{\pi }{4}\end{array}$$

  
  ::A=12r2A=12(1)(1)22A4

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  Area of the sector of doughnut:

  $$\begin{array}{rl}A& =\frac{9\pi }{4}-\frac{\pi }{4}=\frac{8\pi }{4}=2\pi \end{array}$$

  
  ::甜甜圈部门面积:A=944=84=2

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  3. A driver is traveling around a circular track that has radius of 70 meters. If the angle from the starting line to her current position is $\begin{array}{r}\frac{\pi }{3}\end{array}$ radians, what is the area of the sector traced out by her car?
  ::3. 驾驶员环绕圆形轨道行驶,半径70米,如果从起点到目前位置的角是 3 弧度,则其车辆所追查到的分区面积是多少?

  

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  The area of a sector is:

  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \end{array}$$

  . This leads us to:
  ::A=12r2。这导致我们:

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}(70{)}^2\frac{\pi }{3}\\ A& \approx 73.3\end{array}$$

  meters
  ::A=12r2A=12(70)23A=73.3米

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked what is the area of your pie piece. 
  ::之前有人问过你,你的馅饼饼是什么区域?

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  You now know that the equation for the area of a circle swept out by some angle is:
  ::你们现在知道圆圈区域以某种角度分布的方程式是:

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  $$\begin{array}{r}A=\frac{1}{2}{r}^2\theta \end{array}$$

  
  ::A=12r2

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  Applying this to your pie slice, you have an area of:
  ::把这个应用到你的馅饼切片上, 你有一个区域, 包括:

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  $$\begin{array}{r}A=\frac{1}{2}\times 6^2\times \frac{\pi }{6}=3\pi \end{array}$$

  
  ::A=12x626=3

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  And the total area of the pie is:
  ::馅饼的总面积是:

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  $\begin{array}{r}\pi r^2=36\pi \end{array}$
  ::°r2=36_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  To find out what your fair portion of the pie is, multiply the total area by your fraction:
  ::为了找出你那部分派的分量, 将总面积乘以分数 :

  $\begin{array}{r}\frac{1}{5}\times 36\pi =7.2\pi \end{array}$

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  And since the piece you took is only $\begin{array}{r}3\pi \end{array}$ in size, you are definitely not taking too much!
  ::既然你拿的那块只有3大小 你绝对不会拿太多!

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  Example 2
  ::例2

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  If the radius of a sector is 5 feet, and the sector sweeps out an angle of $\begin{array}{r}{43}^{\circ }\end{array}$ , find the area of the sector.
  ::如果一个部门的半径是5英尺, 而这个部门的角度是43英寸, 找到这个部门的面积。

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  Since you know that $\begin{array}{r}A=\frac{1}{2}{r}^2\theta \end{array}$ , you can solve for the area (don't forget to convert the degrees of the angle to radians):
  ::由于您知道 A=12r2, 您可以为区域解析( 不要忘记将角度的度转换为弧度) :

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  $$\begin{array}{r}A=\frac{1}{2}{r}^2\theta \\ A=\frac{1}{2}(25)(.75)\\ A=9.375\end{array}$$

  
  ::A=12r2A=12(25)(75)A=9.375

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  Example 3
  ::例3

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  If a pie wedge has an area of 15 square in inches, and the pie has a radius of 9 inches, find the angle swept out by the sector.
  ::如果馅饼的面积是15平方英寸, 而馅饼的半径是9英寸,

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  Since you know that $\begin{array}{r}A=\frac{1}{2}{r}^2\theta \end{array}$ , you can solve for the angle swept out by the sector (don't forget that the angle will be measured in radians):
  ::由于您知道 A=12r2, 您可以解答扇区冲出的角度( 不要忘记角度将以弧度测量 ) :

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  $$\begin{array}{r}A=\frac{1}{2}{r}^2\theta \\ 15=\frac{1}{2}(81)(\theta )\\ \theta =\frac{(2)(15)}{81}\\ \theta =.37\end{array}$$

  
  ::A=12r215=12(81)(____) (2)(15)8137

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  Example 4
  ::例4

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  If you have a piece of round cake that has an area of 20 square inches, and you know the piece sweeps out an angle of $\begin{array}{r}{25}^{\circ }\end{array}$ , find the radius of the cake.
  ::如果你有一个圆形蛋糕, 面积为20平方英寸, 你知道那块蛋糕扫出角角是25英寸, 找到蛋糕的半径。

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  Since you know that $\begin{array}{r}A=\frac{1}{2}{r}^2\theta \end{array}$ , you can solve for the radius of the cake (don't forget to convert the degrees of the angle to radians):
  ::由于您知道 A=12r2, 您可以解析蛋糕的半径( 不要忘记将角度的度转换为弧度) :

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  $$\begin{array}{r}A=\frac{1}{2}{r}^2\theta \\ 20=\frac{1}{2}(r^2)(.436)\\ r^2=\frac{(2)(20)}{.436}\\ r^2=91.743\\ r=9.58\end{array}$$

  
  ::A=12r220=12(r2(436)r2=(2)(20).436r2=91.743r=9.58

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  Review
  ::回顾

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  1. If the radius of a sector is 8 inches, and the central angle of the sector is $\begin{array}{r}{40}^{\circ }\end{array}$ , find the area of the sector.
     ::如果一个部门的半径是8英寸,而这个部门的中心角是40英寸,请找到这个部门的面积。
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  2. If the radius of a sector is 12 inches, and the central angle of the sector is $\begin{array}{r}\frac{\pi }{6}\end{array}$ radians, find the area of the sector.
     ::如果一个部门的半径是12英寸,而该部门的中心角是 6 弧度,则找到该部门的面积。
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  3. If the radius of a sector is 6 inches, and the central angle of the sector is $\begin{array}{r}{140}^{\circ }\end{array}$ , find the area of the sector.
     ::如果一个部门的半径是6英寸,而这个部门的中心角是140英寸,请找到这个部门的面积。
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  4. If the radius of a sector is 5 inches, and the central angle of the sector is $\begin{array}{r}\frac{5\pi }{3}\end{array}$ radians, find the area of the sector.
     ::如果一个部门的半径是5英寸,而这个部门的中心角是5-3弧度,则找到这个部门的面积。
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  5. If the radius of a sector is 10 inches, and the central angle of the sector is $\begin{array}{r}{100}^{\circ }\end{array}$ , find the area of the sector.
     ::如果一个部门的半径是10英寸,而这个部门的中心角是100英寸,请找到这个部门的面积。
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  6. If a pie wedge has an area of 10 square inches, and the pie has a radius of 6 inches, find the angle swept out by the sector.
     ::如果馅饼的面积是10平方英寸, 而馅饼的半径是6英寸,
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  7. If a pie wedge has an area of 15 square inches, and the pie has a radius of 4 inches, find the angle swept out by the sector.
     ::如果馅饼的面积是15平方英寸, 而馅饼的半径是4英寸,
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  8. If a pie wedge has an area of 12 square inches, and the pie has a radius of 3 inches, find the angle swept out by the sector.
     ::如果馅饼的面积是12平方英寸, 而馅饼的半径是3英寸,
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  9. If you have a piece of round cake that has an area of 20 square inches, and you know the piece sweeps out an angle of $\begin{array}{r}\frac{\pi }{3}\end{array}$ radians, find the radius of the cake.
     ::如果你有一块圆饼, 面积为20平方英寸, 你知道那块蛋糕扫出 3 弧度角, 找到蛋糕的半径。
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  10. If you have a piece of round cake that has an area of 100 square inches, and you know the piece sweeps out an angle of $\begin{array}{r}{50}^{\circ }\end{array}$ , find the radius of the cake.
      ::如果你有一块圆饼, 面积100平方英寸, 你知道那块蛋糕扫出50角角, 找到蛋糕的半径。
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  11. If you have a piece of round cake that has an area of 35 square inches, and you know the piece sweeps out an angle of $\begin{array}{r}\frac{2\pi }{5}\end{array}$ radians, find the radius of the cake.
      ::如果你有一块圆饼, 面积35平方英寸, 你知道这块蛋糕扫出角角 2°5 弧度, 找到蛋糕的半径。
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  12. If you have a piece of round cake that has an area of 20 square inches, and you know the piece sweeps out an angle of $\begin{array}{r}{30}^{\circ }\end{array}$ , find the radius of the cake.
      ::如果你有一块圆饼, 面积为20平方英寸, 你知道那块蛋糕扫出角角30英寸, 找到蛋糕的半径。

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  A pizza has a radius of 10in. Use this information to answer questions 13-15.
  ::披萨半径为10英寸 用这个信息回答问题13-15

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  13. A slice is removed. The length of the crust of the missing slice is 3in. What is the area of the missing slice?
      ::切片被切除。 遗漏切片的地壳长度是 3 英寸。 遗漏切片的面积是 多少 ?
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  14. You eat three pieces with a central angle of $\begin{array}{r}\frac{4\pi }{5}\end{array}$ . What is the area of the pizza you ate?
      ::你吃三片,中央角度是4 5,你吃披萨的面积是多少?
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  15. A large pizza has a radius of 12in. What is the area of half of the large pizza?
      ::大披萨的半径是12英寸 大披萨的一半面积是多少?

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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 。