8.10 区区-interactive

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  区区

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  If a circle with a marked radius is turned a bit, the area enclosed by the two radii looks like a pizza slice or a wedge of pie, this is called a sector of the circle. Hence, a sector of a circle is a region bounded by an arc of the circle and the two radii to the endpoints of the arc, where the smaller area (shaded portion) is known as the minor sector and the larger being the major sector .
  ::如果一个带有标记半径的圆圈被转小一点,则两个半径所包围的区域看起来像一个披萨切片或馅饼,这称为圆圈的一个区块,因此,一个圆圈的一个区块是圆弧的区域,两个半径与弧的终点交界,较小的区域(阴影部分)被称为小区块,较大的区域是主要区块。

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  Because a sector is two dimensional, you can calculate its area. The area of a whole circle with radius  $\begin{array}{r}r\end{array}$ is $\begin{array}{r}\pi r^2\end{array}$ . The area of a sector represents a fraction of this whole circle area . The measure of the central angle helps to tell you what fraction of the circle the sector is. In the problems below, you will show that $\begin{array}{r}\text{Sector Area}=\frac{r^2\theta }{2}\end{array}$ , where  $\begin{array}{r}r\end{array}$ is the radius of the circle and  $\begin{array}{r}\theta \end{array}$ is the measure of the central angle in radians.
  ::由于一个扇区是二维的, 您可以计算其区域。 一个半径为 r 的整圆区域是 r2 。 一个扇区的面积代表整个圆区的一部分。 中心角的测量有助于告诉您该扇区是哪一部分的圆。 在下面的问题中, 您将显示“ 扇区”= r2, 其中 r 是圆的半径, 是弧度中角的测量值。

  

  INTERACTIVE

  Sector Area

  

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  Move the red point to change the angle of the sector. Notice that when the sector has an angle measure of ${360}^{\circ}$  the sector becomes a circle.
  ::移动红点以改变扇区角度 。 请注意, 当扇区角度测量为 360 时, 扇区会变成圆 。

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  Let's take a look at some problems about sector area.
  ::让我们来研究一下部门领域的一些问题。

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  1. In terms of $\begin{array}{r}\theta \end{array}$ , what fraction of the circle is the shaded sector ? Assume  $\begin{array}{r}\theta \end{array}$ is in radians.
  ::1. 就___而言,阴影区是圆的哪一部分?___以弧度表示。

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  Thinking in radians, a full circle is  $\begin{array}{r}2\pi \end{array}$ radians. The sector is therefore  $\begin{array}{r}\frac{\theta }{2\pi }\end{array}$ of the whole circle.
  ::以弧度计,整个圆圈为 2 弧度。 因此, 扇区是整个圆圈的 2 。

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  2. Explain why the area of the sector below is  $\begin{array}{r}\frac{r^2\theta }{2}\end{array}$ where  $\begin{array}{r}\theta \end{array}$ is the measure of the central angle in radians.
  ::2. 解释为何以下部门面积为r22, 其中是弧度中角的测量值。

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  The area of the whole circle is  $\begin{array}{r}\pi r^2\end{array}$ and this sector represents  $\begin{array}{r}\frac{\theta }{2\pi }\end{array}$ of the whole circle. Therefore, the area of the sector is:
  ::整个圆圈的面积为 °r2, 这个扇区代表整个圆圈的 2 。 因此, 这个圆圈的面积是 :

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  $\begin{array}{r}\pi r^2\cdot \frac{\theta }{2\pi }=\frac{r^2\theta }{2}\end{array}$
  ::22222222222222222222222222222222222222222222222

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  3. Find the area of the shaded sector below.
  ::3. 找出下面的阴影区。

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  The angle is given in degrees, so first convert it to radians. Remember that $\begin{array}{r}1\text{degree}=\left(\frac{\pi }{180}\right)\text{radians}\end{array}$ . This means that $\begin{array}{r}{64}^{\circ }=64\cdot \left(\frac{\pi }{180}\right)\approx 1.12\text{radians}\end{array}$
  ::角度以度表示, 所以先将其转换为弧度 。 记住 1 度= (180) radians 。 这意味着 64 64 ( 180) 1. 12 弧度

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  $\begin{array}{r}\text{Sector Area}=\frac{r^2\theta }{2}\approx \frac{(3^2)(1.12)}{2}\approx 5.03{\text{in}}^2\end{array}$
  ::区=r22(32) (1.12) 5.03英寸

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  Sector Area
  ::区区

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  Play with the circle by dragging the points that define the endpoints of the sector. The sector area is recalculated as you drag. 
  ::拖曳点以定义扇区端点。 扇区区域会随着拖动重新计算 。

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  Examples
  ::实例实例实例实例

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  Example 1
  ::例1

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  You make a 10″ diameter cheesecake for a party and coat the top in a layer of chocolate. You divide the cheesecake up into equal slices so that the central angle of each slice is $\begin{array}{r}{15}^{\circ }\end{array}$ . How many square inches of chocolate are on each slice?
  ::你为聚会做一个直径 10 直径的芝士蛋糕, 把顶部涂上一层巧克力。 你将芝士蛋糕分成相等的切片, 这样每个切片的中心角是 15 。 每切片上有多少平方英寸的巧克力?

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  The top of each slice covered in chocolate is a sector with radius 5 inches and a central angle of $\begin{array}{r}{15}^{\circ }\end{array}$ . To find the area of the sector, first convert  $\begin{array}{r}{15}^{\circ }\end{array}$ to radians. Remember that $\begin{array}{r}1\text{degree}=\left(\frac{\pi }{180}\right)\text{radians}\end{array}$ . This means that $\begin{array}{r}{15}^{\circ }=15\cdot \left(\frac{\pi }{180}\right)\approx 0.26\text{radians}\end{array}$ .
  ::巧克力覆盖的每个切片的顶部是半径为5英寸、中心角为15英寸的区块。 要找到扇形区域, 请先将 15 英寸转换为弧度 。 请记住 1 度= (180) 弧度。 这意味着 15 15 ( 180) 0. 26 弧度 。

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  $\begin{array}{r}\text{Chocolate Area}=\frac{r^2\theta }{2}\approx \frac{(5^2)(0.26)}{2}\approx 3.27{\text{in}}^2\end{array}$
  ::巧克力面积=r22(52)(0.262) 3.27英寸 2

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  Example 2
  ::例2

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  Find the area of the sector below:
  ::部门面积如下:

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  To find the area of the sector, first convert  $\begin{array}{r}{100}^{\circ }\end{array}$ to radians. Remember that $\begin{array}{r}1\text{degree}=\left(\frac{\pi }{180}\right)\text{radians}\end{array}$ . This means that $\begin{array}{r}{100}^{\circ }=100\cdot \left(\frac{\pi }{180}\right)\approx 1.75\text{radians}\end{array}$ .
  ::要找到扇区区域, 请先将 100 转换为弧度 。 请记住 1 度= (180) 弧度。 这意味着 100 100 (180) 1. 75 弧度 。

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  $\begin{array}{r}\text{Sector Area}=\frac{r^2\theta }{2}\approx \frac{(7^2)(1.75)}{2}\approx 42.76{\text{cm}}^2\end{array}$
  ::地区=r22(72)(1.752)42.76平方厘米

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  Example 3
  ::例3

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  Find the area of the triangle below:
  ::查找三角形区域如下:

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  Because the radius of the circle is 7 cm, two sides of the triangle are length 7 cm.
  ::因为圆的半径是7厘米,三角形的两边是7厘米。

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  You know two sides and an included angle so you can find the area using the sine area formula.
  ::您知道两个侧面和一个包含角度, 这样您就可以使用正弦区域公式找到区域 。

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  $\begin{array}{rl}\text{Triangle Area}& =\frac{1}{2}ab\sin C\\ \text{Triangle Area}& =\frac{1}{2}(7)(7)\sin ({100}^{\circ })\\ \text{Triangle Area}& =\frac{1}{2}(7)(7)(0.985)\\ \text{Triangle Area}& \approx 24.12{\text{cm}}^2\end{array}$
  ::三角区域=12absinCTriangle区域=12(7)(7)sin(100) 三角区域=12(7)(7)(7)(0.985) 三角区域=24.12厘米

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  Example 4
  ::例4

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  Use your answers to Example #2 and #3 to find the area of the circular segment below.
  ::使用您对例2和例3的回答来查找以下圆环段的区域 。

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  The circular segment is the portion of the sector not included in the triangle. To find its area, subtract the area of the triangle from the area of the sector.
  ::圆环段是三角形中未包含的部分。要找到三角形区域,请从三角形区域中减去三角形区域。

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  $\begin{array}{r}\text{Segment Area}=\text{Sector Area}-\text{Triangle Area}\end{array}$ $\begin{array}{r}=42.76{\text{cm}}^2-24.12{\text{cm}}^2=18.64{\text{cm}}^2\end{array}$
  ::片段 区域= 区 - 区 - 三角区域= 42.76 cm2- 24.12 cm2= 18.64 cm2

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  CK-12 PLIX Interactive
  ::CK-12 PLIX 交互式互动

  Summary

  播放段落 A sector of a circle is a portion of a circle contained between two radii of the circle. Sectors can be measured in degrees. ::一个圆的扇区是圆两半之间包含的圆的一部分。可以以度度测量扇区。 播放段落 The area of a sector can be found with: $\begin{array}{r}A=\frac{r^2\theta }{2}\end{array}$   ::A=r22 可找到一个部门区域 。 播放段落 Minor sector has a central angle is less than 180 o . ::小部门的中心角小于180o。 播放段落 Major sector has a central angle is more than 180 o . ::主要部门的中心角度超过180o。

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  Review
  ::审查审查审查审查

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  1. Explain why the area of a sector is  $\begin{array}{r}\frac{r^2\theta }{2}\end{array}$ where  $\begin{array}{r}\theta \end{array}$ is the measure of the central angle in radians.
  ::1. 解释为什么一个部门的面积为r22,其中 是弧度中角的测量。

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  2. How do you find the area of a sector if the central angle is given in degrees?
  ::2. 如果中角以度表示,您如何找到某一区的面积?

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  Find the area of each region shaded in blue.
  ::找到每个区域以蓝色阴影的面积。

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  16. A lawn sprinkler is set in the corner of the backyard. Thomas has the sprinkler set to rotate at 90 degrees. He knows that the sprinkler will spray 30 feet. If his backyard is 850 square feet, does Thomas have the sprinkler set to reach most of his yard? Explain your reasoning and any suggestions you might have for Thomas if you find he does not have the sprinkler set properly.
  ::16. 后院角落设置了洒草机,Thomas将喷洒机设定在90度旋转,他知道喷洒机将喷洒30英尺,如果他的后院是850平方英尺,Thomas是否将喷洒机设定在院子里的大部分地方?解释你的理由和任何建议,如果发现Thomas没有适当的喷洒机,你可能对他有什么建议。

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  17. A windshield wiper usually rotates between 80 and 95 degrees if your car uses only one wiper. You are deciding on switching to a more sporty 7 inch wiper from your current 10 inch wiper. Is this a good idea? Explain your reasoning.
  ::17. 如果汽车只使用一个擦拭器,挡风玻璃擦拭器通常在80至95度之间旋转。您决定从目前的10英寸擦拭器中改用一个更运动的7英寸擦拭器。这是否是一个好主意?请解释你的理由。

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  Review (Answers)
  ::审查(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 。