10.10涉及圆的组合图形的面积

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• 选择节涉及圆的组合图形的面积

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  涉及圆的组合图形的面积

  

  [Figure 1]

   

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  Fred’s hobby is wood working, and today he is building a dollhouse for his daughter. The house will include two front bay windows with the  dimensions  below. Fred needs to know the  area  of the window, so he can have the plastic windows cut at the crafts store.
  ::弗雷德的爱好是木材工作,而今天他正在为女儿建造一个娃娃屋。 房子将包括两个前海湾窗口,其尺寸如下。 弗雷德需要了解窗口的面积,以便可以在手工艺品店切下塑料窗户。

  

  [Figure 2]

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  How can Fred determine the area of the window?
  ::Fred如何确定窗口的面积?

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  In this concept, you will learn to find areas of combined figures involving parts of circles.
  ::在这个概念中,你将学会找到包含部分圈子的综合数字领域。

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  Finding Areas of Combined Figures
  ::综合数字查找领域

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  Sometimes, there will be figures that are not quadrilaterals or circles, but are combined figures. A  combined figure  is a figure that is made up of more than one type of  polygon . You can still figure out the area of combined figures, by finding the area of each of the figures and then combing them to get the total area.
  ::有时,会有一些数字不是四边形或圆形,而是合并的数字。一个合并的数字是由多种多边形组成的数字。你仍然可以找到每个数字的面积,然后对它们进行梳理,以获得全部区域。

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

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  Take a look at the figure below. It is a half  circle  and a  rectangle  combined into one figure.
  ::看看下图。它是一个半圆形和一个矩形组合成一个图。

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  What is the area of the figure?
  ::数字的方面是什么?

  

  [Figure 3]

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  To solve this problem, you first have to look at which figures have been combined. Here you have one-half of a circle and a rectangle.
  ::要解决这个问题, 您必须首先查看哪些数字是合并的。 您有半个圆形和一个矩形 。

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  You will need to figure out the area of the rectangle, the area of half of the circle, and then add the two areas. This will give you the area of the combined figure.
  ::您需要找到矩形的面积, 圆半的面积, 然后添加这两个区域。 这将为您提供组合图的面积 。

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  Let’s start with the rectangle.
  ::让我们从矩形开始。

  $$\begin{array}{r}A=lw\end{array}$$

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  The length of the rectangle is 6 inches. The width of the rectangle is 3 inches.
  ::矩形的长度为 6 英寸。矩形的宽度为 3 英寸。

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  First, substitute the given information into the equation to find the area of the rectangle.
  ::首先,将给定信息替换为方程,以找到矩形区域。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (6)(3)\\ A& =& 18i{n}^2\end{array}\end{array}$$

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  The rectangle’s area is  $\begin{array}{r}18i{n}^2\end{array}$ .
  ::矩形区域为 18 i n 2 。

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  Next, find the area of the circle.
  ::下一个,找到圆圈的区域。

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  Notice that the length of the rectangle is also the  diameter  of the circle, which is 6 inches. If the diameter of the circle is 6 inches, then the  radius  is 3 inches. Remember that the radius is one-half of the diameter.
  ::请注意矩形的长度也是圆的直径,即6英寸。如果圆的直径是6英寸,则半径是3英寸。请记住半径是直径的半径。

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  The goal is to figure out the area of one whole circle and then divide that in half for the area of half of the circle.
  ::目标是找出整个圆圈的面积,然后将半圆圈的面积分成一半。

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  First, substitute the given information into the equation for the area of a circle.
  ::首先,将给定信息替换为圆区域方程。

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

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  This is the area of the whole circle.
  ::这是整个圆圈的区域。

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  Next, because the figure is only half of a circle, divide this in two.
  ::接下来,因为这个数字只有半个圆圈, 将其分成两个。

  $$\begin{array}{r}28.26÷2=14.13in\end{array}$$

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  This is the area of the half circle.
  ::这是半圆的面积

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  Then, combine the area of the rectangle with the area of the half circle. This will equal the area of the entire figure.
  ::然后,将矩形区域与半圆区域结合起来。这将等于整个图的面积。

  $$\begin{array}{r}18+14.13=32.13\end{array}$$

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  The area of the combined figure is  $\begin{array}{r}32.13i{n}^2\end{array}$ .
  ::合并数字的面积为32.13 i n 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 Fred and the dollhouse he’s building for his daughter.
  ::更早之前, 你得到一个关于Fred和他为女儿建造的娃娃屋的问题。

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  Fred designed two bay windows that were a combination of a rectangle and a half circle. The windows have the following dimensions.
  ::Fred设计了两个海湾窗口,它们是一个矩形和半圆的组合。这些窗口有以下尺寸。

  

  [Figure 4]

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  Fred knows that the total area of the window will be the combined area of the rectangle and the half circle, so he decides to find the area of each and then combine them.
  ::Fred知道窗口的总面积将是矩形和半圆的合并区域,所以他决定找到每个区域,然后将其合并。

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  First, Fred finds the area of the rectangle by substituting the given information into the equation.
  ::首先,Fred找到矩形区域, 将给定的信息替换为方程式 。

  $$\begin{array}{r}A=lw\end{array}$$

  The length is 6 inches and the width is 8 inches.

  $$\begin{array}{r}A=(6)(8)=48\text{sq}.\text{inches}\end{array}$$

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  The area of the rectangle is 48  square  inches.
  ::矩形区域为48平方英寸。

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  Next, this is a whole circle divided in half, so Fred finds the area of the whole circle by substituting the given information into the equation. Note that the radius is 4, half of the 8-inch diameter of the circle.
  ::接下来,这是一个整圆, 半分为一半, 所以Fred会找到整个圆的区域, 将给定的信息替换为方程。 请注意, 半径是 圆的 8 英寸直径的 4, 半径 。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)r^2\\ A& =& (3.14)4^2\\ A& =& (3.14)(16)\\ A& =& 50.24\text{sq}.\text{inches}\end{array}\end{array}$$

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  Then, because this is the area of the whole circle, divide it by 2.
  ::然后,由于这是整个圆圈的面积, 将其除以2。

  $$\begin{array}{r}50.24÷2=25.12\text{sq}.\text{inches}\end{array}$$

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  The area of the half circle is 25.12 square inches.
  ::半圆面积为25.12平方英寸。

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  Finally, add the area of the rectangle with the area of the half circle.
  ::最后,加上半圆的矩形区域。

  $$\begin{array}{r}48+25.12=73.12\text{sq}.\text{inches}\end{array}$$

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  The area of the combined figure is 73.12 square inches. So, each bay window has an area of 73.12 square inches.
  ::合并数字的面积为73.12平方英寸。因此,每个海湾窗口的面积为73.12平方英寸。

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  Example 2
  ::例2

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  Find the area of the combined figure below.
  ::请在下方查找合并图的面积。

  

  [Figure 5]

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  First, find the area of the rectangle by substituting the given information into the equation.
  ::首先,找到矩形区域, 将给定的信息替换为方程式 。

  $$\begin{array}{r}A=lw\end{array}$$

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  The length is 24 ft. and the width is 8 feet.
  ::长度为24英尺,宽度为8英尺。

  $$\begin{array}{r}A=(8)(24)=192\text{sq}.\text{feet}\end{array}$$

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  The area of the rectangle is  $\begin{array}{r}192\text{sq}.\text{feet}\end{array}$ .
  ::矩形区域为192平方英尺。

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  Next, find the area of the circle as if it were a whole circle by substituting the given information into the equation.
  ::下一步,找到圆区域,将给定的信息换成方程式,以发现圆区域仿佛是一个整圆。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)r^2\\ A& =& (3.14){12}^2\\ A& =& (3.14)(144)\\ A& =& 452.16\text{sq}.\text{feet}\end{array}\end{array}$$

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  Then, divide the area of the whole circle in half.
  ::然后,将整个圆的面积一分为二。

  $$\begin{array}{r}A=226.08\text{sq}.\text{feet}\end{array}$$

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  Finally, add the area of the rectangle with the area of half of the circle.
  ::最后,加上圆半的矩形区域。

  $$\begin{array}{r}192+226.08=418.08\text{sq}.\text{feet}\end{array}$$

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  The area of the combined figure is 418.08 sq. feet.
  ::合并数字的面积为418.08平方英尺。

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  Example 3
  ::例3

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  Find the area of the combined figure. Remember, separate the figure and find the area of the parts, then combine the areas.
  ::查找合并图的区域。 记住, 分隔图, 找到部分的区域, 然后合并区域 。

  

  [Figure 6]

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  First, find the area of the square in the middle by substituting the given information into the equation.
  ::首先,找到中间方块的面积,将给定的信息替换为方程式。

  $$\begin{array}{r}A=lw\end{array}$$

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  The length is 6 inches and the width is 6 inches.
  ::长度为6英寸,宽度为6英寸。

  $$\begin{array}{r}A=(6)(6)=36\text{sq}.\text{inches}\end{array}$$

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  The area of the square is 36 square inches.
  ::广场面积为36平方英寸。

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  Next, this is a whole circle divided into two by the square, so find the area of the whole circle by substituting the given information into the equation. Note that the radius is 3 in.
  ::接下来,这是整个圆形,按正方形分为两个,然后以给定信息替换方程,找到整个圆形的区域。请注意半径是3英寸。

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

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  This is the combined area of both halves of the circle.
  ::这是圆两半的合并区域。

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  Finally, add the area of the rectangle with the area of the circle.
  ::最后,加上圆区域之间的矩形区域。

  $$\begin{array}{r}36+28.26=64.26\text{sq}.\text{inches}\end{array}$$

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  The area of the combined figure is 64.26 square inches.
  ::合并数字的面积为64.26平方英寸。

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  Example 4
  ::例4

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  Answer the following question.
  ::回答下列问题。

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  Can you figure out the area of a figure made up of two  congruent circles ? How?
  ::你能找出由两个相似的圆圈组成的数字区域吗?

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  First, you could figure out the area of both circles using the given information.
  ::首先,你可以使用给定的信息 找出两个圈子的面积。

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  Next, you would add the areas of both circles to get the area of the combined figure.
  ::接下来,你将添加两个圈子的区块 以获得合并数字的区块。

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  Example 5
  ::例5

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  Find the area of the figure.
  ::找到数字的区域 。

  

  [Figure 7]

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  This is a half circle.
  ::这是一个半圆。

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  First, note that the radius is half of the 17-inch diameter. So the radius is 8.5 inches.
  ::首先,请注意半径为 17 英寸直径的一半。 所以半径是 8.5 英寸。

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  Next, substitute this information into the area of a circle equation.
  ::接下来,将此信息替换为圆形方程式区域。

  $$\begin{array}{r}\begin{array}{rcl}A& =& (3.14)r^2\\ A& =& (3.14){8.5}^2\\ A& =& (3.14)(72.25)\\ A& =& 226.87\text{sq}.\text{inches}\end{array}\end{array}$$

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  This is the area of the whole circle.
  ::这是整个圆圈的区域。

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  Then, divide the area of the whole circle by 2 to get the area of the figure.
  ::然后将整个圆的面积除以 2 以获得数字的区域 。

  $$\begin{array}{r}226.87÷2=113.43\text{sq}.\text{inches}\end{array}$$

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  The area of the figure is 113.43 square inches.
  
  ::图中面积为113.43平方英寸。

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  Review
  ::回顾

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  Use each image to answer the following questions. Use the value approximated value 3.14 for the number  $\begin{array}{r}\pi .\end{array}$  Round your answer to the nearest hundredths place if needed.
  ::使用每个图像回答下列问题。 使用数字 的约合值 3. 14 值。 如果需要, 将您的答复四舍五入到最接近的百位 。

  

  [Figure 8]

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  1. Which two figures are pictured here?
     ::这里所描绘的是哪两个数字?
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  2. What is the area of the rectangle?
     ::矩形的区域是什么?
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  3. What is the area of the circle if it were a whole circle?
     ::如果圆是一个整圈,圆的面积是什么?
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  4. What is the area of half of the circle?
     ::半圆的面积是多少?
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  5. What is the area of the whole figure?
     ::整个数字的方面是什么?

  

  [Figure 9]

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  6. What is the radius of this circle?
     ::这个圆的半径是多少?
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  7. What is the diameter of this circle?
     ::这个圆的直径是多少?
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  8. What is the circumference of this circle?
     ::这个圆圈的环形是什么?
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  9. What is the area of the circle?
     ::圆圈的区域是什么?
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  10. If this circle were half of a circle, what would the new area be?
      ::如果这个圆圈是半圆圈,新区域会是什么?

  

  [Figure 10]

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  11. What is the radius of this circle?
      ::这个圆的半径是多少?
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  12. What is the diameter of this circle?
      ::这个圆的直径是多少?
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  13. What is the circumference of this circle?
      ::这个圆圈的环形是什么?
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  14. What is the area of the circle?
      ::圆圈的区域是什么?
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  15. If this circle were half of a circle, what would the new area be?
      ::如果这个圆圈是半圆圈,新区域会是什么?

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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