2.5正方形和矩形的面积

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  正方形和矩形的面积

   

  

  [Figure 1]

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  Allen bought a new house and is planning to replace the carpet in two of the bedrooms. One bedroom is a shaped like a  square  and the other is rectangular. He needs to figure out how much carpet is needed to cover both bedroom floors.
  ::艾伦买了一栋新房子,并计划在两个卧室中更换地毯。一个卧室的形状像一个广场,另一个是长方形。他需要弄清楚需要多少地毯来覆盖卧室的两层楼。

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  The square bedroom has side lengths of 9 feet.
  ::方形卧室的侧长为9英尺

  

  [Figure 2]

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  The rectangular bedroom has a length of 12 feet and a width of 8 feet.
  ::矩形卧室长12英尺,宽8英尺。

  

  [Figure 3]

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  In order for Allen to figure out how much carpet he needs to buy for these two bedrooms, he needs to find the  area  of each bedroom. 
  ::为了让艾伦弄清楚他需要买多少地毯 买给这两间卧室 他需要找到每个卧室的面积

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  In this concept, you will learn how to find the area of squares and rectangles.
  ::在这个概念中,你会学会如何找到方形和矩形区域。

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  Area
  ::地区面积

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  The inside  space  of a figure is known as the  area  of the figure. Carpeting a floor, grass on the ground, or anything else covering the space inside of a figure are all examples of area.
  ::一个图的内空称为图的面积。 地毯地板、 草地或覆盖图内空间的其他任何东西都是区域的例子 。

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  There is a formula for finding the area of different shapes. In this concept, this concept will show you how to find the area of both squares and  .
  ::找到不同形状区域的公式。 在这个概念中, 这个概念将显示您如何找到 方块和 。

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  This is the formula used to find the area of a square:
  ::这是用来查找正方形面积的公式 :

  $$\begin{array}{r}A=s\cdot s\end{array}$$

  

  [Figure 4]

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  Remember, the dot is another symbol for multiplication. To figure out the area of the square, multiply one side times the other side. Since all sides in a square are equal, the two numbers will be the same. The unit of measurement is kept in the equation to help you remember what the correct label should be.
  ::记住, 点是乘法的另一个符号。 要找出正方块的面积, 请乘以另一边的一倍。 由于正方块中的所有边都是相等的, 这两个数字将是相同的。 测量单位将保存在方程中, 以便帮助您记住正确的标签应该是什么 。

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  Here is how to use the formula for finding the area of a square:
  ::如何使用公式来找到方形区域 :

  $$\begin{array}{r}A=6ft\cdot 6ft\end{array}$$

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  Now, think of the equation as multiplying two different things; the numbers and the units of measurement.
  ::现在,把方程式想象成乘以两个不同的东西:数字和测量单位。

  $$\begin{array}{rl}A& =6\cdot 6\\ A& =ft\cdot ft\end{array}$$

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  Solve both equations.
  ::解决了两个方程式

  $$\begin{array}{rl}A& =6\times 6=36\\ A& =ft\times ft=f{t}^2\end{array}$$

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  Think back to what you know about exponents. When you multiply two of the same thing together, you can write it in exponential form. Another way to label area is to write square feet (sq. ft.).
  ::回想一下你所知道的关于引言的内容。 当您将相同事物中的两件乘在一起时, 您可以以指数形式写入它。 标签区域的另一个方法是写平方英尺( sq. ft.) 。

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  The answer is 36 sq. ft. or  $\begin{array}{r}36{ft}^2\end{array}$
  ::答案是36平方英尺或36平方英尺 t 2

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  This is the formula used to find the area of a  rectangle :
  ::这是用来查找矩形区域的公式 :

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

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  To find the area of a rectangle, use the measurements for length and width instead of side length. These numbers may be different from each other because not all rectangles have the same side lengths like squares do.
  ::要找到矩形的区域, 请使用长度和宽度的测量, 而不是边长。 这些数字可能不同, 因为并非所有矩形的侧长都和方形相同 。

  

  [Figure 5]

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  This rectangle measures a length of 5 meters and a width of 3 meters. Just like the area formula for a square, multiply those two numbers to find the area of the rectangle. Remember, two variables next to each other in an equation indicate multiplication, just like the dot symbol or traditional multiplication symbol. The symbols will be used interchangeably.
  ::此矩形测量的长度为 5 米,宽度为 3 米。 就像一个正方形的面积公式一样, 将这两个数字乘以找到矩形的区域。 记住, 方程式中两个相邻变量表示乘法, 就像点符号或传统的乘法符号一样。 这些符号将被互换使用 。

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  To find the area of a rectangle, multiply length times width.
  ::要找到矩形区域, 需要乘以长度乘以宽度 。

    

  $$\begin{array}{rl}A& =5m\cdot 3m\\ A& =5\times 3\\ A& =meters\times meters\end{array}$$

         

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  The formula shows 5 meters times 3 meters. Multiply the measurement part (5  $\begin{array}{r}\times \end{array}$  3), then multiply the units of measurement.
  ::公式显示5米乘以3米。乘以测量部分(5×3),再乘以测量单位。

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  The answer is 15 sq. m or  $\begin{array}{r}15m^2\end{array}$ .
  ::答案是15平方米或15米2

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  The label for area of a rectangle is  similar  to the label for area of a square from above. The label can be written in exponential form or written in abbreviation. 
  ::矩形区域的标签与上方方区域的标签相似。标签可以指数形式写成,也可以缩写缩写。

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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 Allen and his new house that he wants to re-carpet.
  ::早些时候,你得到一个问题 关于艾伦和他的新房子 他想要重生。

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  The  dimensions  of Allen's two bedrooms are given in the beginning of the concept. His square bedroom has side lengths of 9 feet. The other room is a rectangle with dimensions of 12 feet by 8 feet. To begin solving this problem, find the area of each of the bedrooms.
  ::Allen的两间卧室的尺寸在概念的开头就给出了。 他的平方卧室的侧长为9英尺。 另一个房间是12英尺乘8英尺的矩形。 为了解决这个问题, 找到每个卧室的面积 。

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  The square bedroom has a side length of 9 feet.
  ::方形卧室的侧长为9英尺

  $$\begin{array}{rl}A& =s\cdot s\\ A& =9\times 9=81sq.feet\end{array}$$

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  The square bedroom has an area of 81 square feet.
  ::广场卧室面积为81平方英尺。

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  The rectangular bedroom has a length of 12 feet and width of 8 feet.
  ::矩形卧室长12英尺,宽8英尺。

  $$\begin{array}{rl}A& =12ft\cdot 8ft\\ A& =12\times 8\\ A& =feet\times feet\end{array}$$

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  The rectangular bedroom has an area of 96 square feet.
  ::矩形卧室面积96平方英尺。

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  Allen wants the total area of his two bedrooms to know how much carpet to buy, so add the two areas together. 
  ::艾伦想要他两间卧室的总面积 来知道要买多少地毯 所以把这两间房间加在一起

  $$\begin{array}{r}81+96=177squarefeet\end{array}$$

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  The answer is 177 square feet.
  ::答案是177平方英尺

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  Example 2
  ::例2

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  Find the area of a rectangle with a length of 7 ft. and a width of 4 ft.
  ::查找长为 7 英尺和宽度为 4 英尺的矩形区域。

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  First, write out the equation 
  ::首先,写出方程式

  $$\begin{array}{rl}A& =lw\\ A& =7ft\cdot 4ft\end{array}$$

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  Next, multiply the two numbers. 
  ::下一个,乘以两个数字。

  $$\begin{array}{r}A=7\cdot 4=28\end{array}$$

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  Then, multiply the units of measurement.
  ::然后,乘以测量单位。

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

   

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  Now, put your answer together to show the final solution.
  ::现在,把答案放在一起 来展示最终的解决方案

  $$\begin{array}{r}A=28{ft}^2\end{array}$$

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  The answer is 28 sq. ft.
  ::答案是28平方英尺

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  Once you do more examples you may find yourself skipping the step of multiplying the units of measurement. This may become natural to square the unit of measurement for any area problem.
  ::一旦您做了更多的例子, 你可能会发现自己跳过乘以计量单位的步骤。 这可能会自然成为任何区域问题的平方度量单位。

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  Example 3
  ::例3

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  Find the area of a square with side lengths of 10 inches.
  ::找到一个平方的区域,侧长10英寸。

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  First, write your equation.
  ::首先,写你的方程式

  $$\begin{array}{rl}A& =s\cdot s\\ A& =10in\cdot 10in\end{array}$$

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  Next, solve the equation by multiplying the two side lengths.
  ::接下来,通过乘以两个侧边长度来解析方程。

   

  $$\begin{array}{r}A=10\cdot 10=100\end{array}$$

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  Then, write your final solution. Remember to write the squared unit of measurement after the numerical answer.
  ::然后,写下最终的解决方案。 记住在数字回答之后写上方形的测量单位 。

  $$\begin{array}{r}A=100{in}^2\end{array}$$

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  The answer is 100 sq. in.
  ::答案是100平方米英寸

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  Example 4
  ::例4

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  Find the area of a rectangle with a length of 6 feet and a width of 5 feet.
  ::找到一个长6英尺宽5英尺的矩形区域。

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  First, write your equation.
  ::首先,写你的方程式

  $$\begin{array}{rl}A& =lw\\ A& =6ft\cdot 5ft\end{array}$$

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  Next, solve the equation by multiplying the length times the width.
  ::接下来,通过乘以宽度的倍数来解析方程式。

  $$\begin{array}{r}A=6\cdot 5=30\end{array}$$

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  Then, write your final solution. Remember to write the squared unit of measurement after the numerical answer.
  ::然后,写下最终的解决方案。 记住在数字回答之后写上方形的测量单位 。

   

  $$\begin{array}{r}A=30{ft}^2\end{array}$$

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  The answer is 30 sq. ft.
  ::答案是30平方英尺

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  Example 5
  ::例5

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  Find the area of a rectangle with a length of 9 meters and a width of 8 meters.
  ::找到一个长9米宽8米的矩形区域。

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  First, write your equation.
  ::首先,写你的方程式

   

  $$\begin{array}{rl}A& =lw\\ A& =9m\cdot 8m\end{array}$$

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  Next, solve the equation by multiplying the length times the width.
  ::接下来,通过乘以宽度的倍数来解析方程式。

   

  $$\begin{array}{r}A=9\cdot 8=72\end{array}$$

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  Then, write your final solution. Remember to write the squared unit of measurement after the numerical answer.
  ::然后,写下最终的解决方案。 记住在数字回答之后写上方形的测量单位 。

  $$\begin{array}{r}A=72m^2\end{array}$$

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  The answer is 72 sq. m
  ::答案是72平方米

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  Review
  ::回顾

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  Find the area of each of the following figures. Make sure you label your answer correctly.
  ::查找下图的每个区域。 请确保您正确标签您的答案 。

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  1. A square with a side length of 6 inches.
     ::边长6英寸的方形
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  2. A square with a side length of 4 inches.
     ::一个侧长4英寸的方形广场。
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  3. A square with a side length of 8 centimeters.
     ::边长8厘米的方形
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  4. A square with a side length of 12 centimeters.
     ::边长12厘米的方形
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  5. A square with a side length of 9 meters.
     ::一个侧长9米的方形。
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  6. A rectangle with a length of 6 inches and a width of 4 inches.
     ::长6英寸长的矩形,宽4英寸。
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  7. A rectangle with a length of 9 meters and a width of 3 meters.
     ::长9米宽3米长的矩形
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  8. A rectangle with a length of 4 meters and a width of 2 meters.
     ::长4米宽2米长的矩形。
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  9. A rectangle with a length of 17 feet and a width of 12 feet.
     ::长17英尺长的矩形 宽12英尺宽
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  10. A rectangle with a length of 22 feet and a width of 18 feet.
      ::长22英尺和宽18英尺的矩形。
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  11. A square with a side length of 13 feet.
      ::边长13英尺的方形
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  12. A square with a side length of 18 feet.
      ::一个侧长18英尺的方形。
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  13. A square with a side length of 21 feet.
      ::一个侧长21英尺的方形。
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  14. A rectangle with a length of 18 feet and a width of 13 feet.
      ::长18英尺宽13英尺的矩形
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  15. A rectangle with a length of 60 feet and a width of 27 feet.
      ::长60英尺宽27英尺的矩形
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  16. A rectangle with a length of 57 feet and a width of 22 feet.
      ::长57英尺宽22英尺的矩形

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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

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