10.4 矩形棱晶表面面积

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• 选择节矩形棱柱表面面积

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  矩形棱柱表面面积

  

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  The workers at a candy factory have been tasked with finding out how much material is required to cover the factory's famous candy bar. The candy bar measures 5 inches long, 2 inches wide, and 1 inch in height .  What size wrapper will be required to cover the candy bar?
  ::一家糖果厂的工人负责查清楚需要多少材料来覆盖该厂著名的糖果棒。 糖果棒的长度为5英寸,2英寸宽,身高为1英寸。 要覆盖糖果棒,需要多少尺寸的包装纸?

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  In this concept, you will learn how to calculate the of a rectangular prism . 
  ::在这个概念中,你会学会如何计算矩形棱柱。

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  Finding the Surface Area of a Rectangular Prism
  ::查找矩形棱柱的表面区域

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  Area is the space that is contained in a two-dimensional figure.  Surface area is the area of all the sides and faces of a three-dimensional figure. Imagine wrapping a in wrapping paper, like a present. The amount of wrapping paper needed to cover the figure represents its surface area.
  ::区域是包含在二维图中的空格。 表面区域是所有侧面和三维图面的面积。 想象用包装纸包起来, 像现在一样。 覆盖该图所需的包装纸数量代表其表面区域 。

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  To find the surface area, you calculate the area of each face or side and then add these areas together.
  ::为了找到表面积,您计算每个面或侧面的面积,然后将这些区域加在一起。

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  One way to do this is to use a , since a net is a picture of an unfolded solid, representing each of its faces and/or sides. Do this by finding the area of each face or side of the net, and then adding all of the areas together.
  ::其中一个方法就是使用一个, 因为网是一张展示出来的固体的图片, 代表每个面部和/或侧面。 找到网的每个面部或侧面的面积, 然后将所有区域加在一起, 来做到这一点 。

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  You can use this process with prisms of all different kinds. Let’s look at a net for a rectangular prism.
  ::您可以使用各种棱镜来使用这个过程。让我们看看长方形棱镜的网吧。

  

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  The area of each face of the rectangular prism above is calculated using the formula  $\begin{array}{r}A=lw\end{array}$ , where A = area, l = length, and w = width.  The unit of measurement for area is square units , in this case  $\begin{array}{r}{inches}^2\end{array}$ .  Next you find the surface area by adding the area of each of the faces together.  The surface area for the above rectangular prism net is 158   $\begin{array}{r}{in}^2\end{array}$   .  
  ::以上矩形棱柱的每个面的面积使用公式A=lw计算,其中A=区域,l=长度,w=宽度。区域测量单位为平方单位,在此情况下为英寸2;接下来,通过将每个面的面积加在一起,找到表面积。上述矩形棱柱网的表面积为158英寸2。

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  Nets let you see each face so that you can calculate their areas. However, you can also use a formula to represent the faces as you find their areas. Let’s look again at our calculations for the rectangular prism net above. Notice that the calculations repeat in pairs. This is because every face in a rectangular prism is opposite a face that is congruent . In other words, the top and bottom faces have the same measurements; the two long side faces have the same measurements, and the two short side faces have the same measurements. Therefore, you can create a formula for surface area that gives us a short cut. You simply double each calculation to represent a pair of faces.
  ::网让您看到每张脸,这样您就可以计算其区域。然而,您也可以使用一个公式来代表您找到的区域时的面孔。让我们再看一看上面矩形棱柱网的计算结果。请注意,计算结果重复了对齐。这是因为矩形棱柱的每张脸对面都是相近的面孔。换句话说,上面和下面的测量结果相同;两张长面的测量结果相同,两张短面的测量结果相同。因此,您可以为表面积创建一个公式,给我们一个短截。您只需将每张图纸的计算数字翻一番就可以代表一对面。

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  The formula looks like this:
  ::公式看起来是这样的:

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  $\begin{array}{r}SA=2lw+2lh+2hw\end{array}$ , where SA = surface area, l = length, w = width, and h = height.
  ::SA=2lw+2lh+2hw,其中SA=表面积,l=长度,w=宽度,h=高度。

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  In the rectangular prism net above, l = 8 inches, w = 5 inches, and h = 3 inches. Simply put these numbers into the formula and solve for surface area. Let’s try it.
  ::在以上矩形棱柱网中, l = 8 英寸, w = 5 英寸, h = 3 英寸。简单地把这些数字放在公式中,解决表面区域的问题。让我们试试。

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  $$\begin{array}{rl}SA& =2lw+2lh+2hw\\ SA& =2(8\times 5)+2(8\times 3)+2(3\times 5)\\ SA& =2(40)+2(24)+2(15)\\ SA& =80+48+30\\ SA& =158in{.}^2\end{array}$$

  Again, the surface area of this prism is 158 square inches. This formula just saves a little time by allowing you to calculate the area of pairs of faces at a time.
  ::SA=2lw+2lh+2hwSA=2(8x5)+2(8x3)+2(3x5)SA=2(40)+2(24)+2(15)SA=80+48+30SA=158(2Again),这一棱镜的表面面积为158平方英寸。这个公式只节省了一点时间,允许您一次计算双面区域。

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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 the candy factory workers, who need to figure out how much wrapper is needed to cover their famous candy bar.
  ::早些时候,有人给了你 糖果厂工人一个问题, 他们需要弄清楚 需要多少包装 来覆盖他们著名的糖果棒。

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  The dimensions of the candy bar are   $\begin{array}{r}{5}^{″}L\times 2^{″}W\times 1^{″}H\end{array}$ .
  ::糖果棒的尺寸是 5Lx2Wx1H。

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  Since the candy bar is a rectangular prism, use the surface area formula:
  ::由于糖果棒是长方形棱柱,使用表面积公式:

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  $$\begin{array}{r}SA=2(lw+lh+wh)\end{array}$$

  
  ::SA=2(lw+lh+wh)

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  First, plug the dimensions of the candy bar into the formula for surface area of a rectangular prism and multiply the values within the brackets:
  ::首先,将糖果棒的维度插入矩形棱柱表面的公式,并乘以括号内的值:

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  $$\begin{array}{rl}SA& =2\cdot [5(2)+5(1)+2(1)]\\ SA& =2(10+5+2)\end{array}$$

  
  ::SA=22[5(2)+5(1)+2(1)]SA=2(10+5+2]

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  Next, add the values together that are inside the parentheses;
  ::下一步,将括号内的数值加在一起;

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  $$\begin{array}{rl}SA& =2(10+5+2)\\ SA& =2(17)\end{array}$$

  
  ::SA=2(10+5+2)SA=2(17)

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  Then, multiply the resulting sum by 2:
  ::然后, 乘以结果的总和乘以 2 :

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  $$\begin{array}{rl}SA& =2(17)\\ SA& =34sq.inches\end{array}$$

  
  ::SA=2(1717)SA=34 sq.inches

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  The answer is the candy bar will require 34 square inches to completely cover the candy bar.
  ::答案是糖果棒需要34平方英寸 才能完全覆盖糖果棒

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  Example 2
  ::例2

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  What is the surface area of the figure below?
  ::下图的表面积是多少?

  

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  All of the faces of this prism are , so you can use the formula for finding the surface area of a rectangular prism as follows.
  ::此棱镜的所有面貌都是 , 所以您可以使用公式来查找 矩形棱镜的表面区域 如下 。

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  First, plug the values given above into the surface area formula and multiply the values together within each of the parentheses:  

  $$\begin{array}{rl}SA& =2lw+2lh+2hw\\ SA& =2(21\times 14)+2(21\times 5)+2(5\times 14)\end{array}$$

  
  ::首先,将上面给定的值插入表面积公式,并在每个括号内将数值相乘:SA=2lw+2lh+2hwSA=2(21x14)+2(21x5)+2(5x14)

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  $$\begin{array}{rl}SA& =2(294)+2(105)+2(70)\end{array}$$

  
  ::SA=2(294)+2(105)+2(70)

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  Next, multiply each value by 2:
  ::下一步,将每个值乘以 2 :

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  $$\begin{array}{rl}SA& =588+210+140\end{array}$$

  
  ::SA=588+210+140

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  Then, add the values together for the final answer.  Remember to include the unit of measurement:
  ::然后,为最终答案将数值相加。请记住包含一个计量单位:

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  $$\begin{array}{rl}SA& =938c{m}^2\end{array}$$

  
  ::SA=938厘米2

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  The answer is the rectangular prism has a surface area of 938 square centimeters. 
  ::答案是矩形棱柱表面面积为938平方厘米。

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  Example 3
  ::例3

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  Find the surface area of a rectangular prism with a length of 8 in, width of 4 inches, height of 6 inches.
  ::查找长8英寸、宽4英寸、高度6英寸的矩形棱柱表面面积。

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  First, plug the values given above into the surface area formula and multiply the values together within each of the parentheses:
  ::首先,将上面给定的值插入表面积公式,并在括号中将数值相乘:

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  $$\begin{array}{rl}SA& =2lw+2lh+2hw\\ SA& =2(8\times 4)+2(8\times 6)+2(6\times 4)\end{array}$$

  
  ::SA=2lw+2lh+2hwSA=2(8x4)+2(8x6)+2(6x4)

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  $$\begin{array}{rl}SA& =2(32)+2(48)+2(24)\end{array}$$

  
  ::SA=2(32)+2(48)+2(24)

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  Next, multiply each value by 2:
  ::下一步,将每个值乘以 2 :

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  $$\begin{array}{rl}SA& =64+96+48\end{array}$$

  
  ::SA=64+96+48

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  Then, add the values together for the final answer.  Remember to include the unit of measurement:
  ::然后,为最终答案将数值相加。请记住包含一个计量单位:

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  $$\begin{array}{rl}SA& =208i{n}^2\end{array}$$

  
  ::SA=208英寸2

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  The answer is the surface area for this rectangular prism is 208 square inches.
  ::答案是这个矩形棱柱的表面面积是208平方英寸。

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  Example 4
  ::例4

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  Find the surface area of a rectangular prism with a length of 5 ft, width of 4 ft, height of 2 ft
  ::查找长5英尺、宽4英尺、高2英尺长的矩形棱晶体表面面积

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  First, plug the values given above into the surface area formula and multiply the values together within each of the parentheses: 

  $$\begin{array}{rl}SA& =2lw+2lh+2hw\\ SA& =2(5\times 4)+2(5\times 2)+2(2\times 4)\end{array}$$

  
  ::首先,将上面给定的值插入表面积公式,并在每个括号内将数值相乘:SA=2lw+2lh+2hwSA=2(5x4)+2(5x2)+2(2x2)+2(2x4)+2(2x4)

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  $$\begin{array}{rl}SA& =2(20)+2(10)+2(8)\end{array}$$

  
  ::SA=2(20)+2(10)+2(8)

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  Next, multiply each value by 2:

  $$\begin{array}{rl}SA& =40+20+16\end{array}$$

  
  ::下一步,将每个值乘以2:SA=40+20+16

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  Then, add the values together for the final answer and include the unit of measurement:

  $$\begin{array}{rl}SA& =76i{n}^2\end{array}$$

  
  ::然后,将最终答案的数值加在一起,并包括计量单位:SA=76 in2。

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  The answer is the surface area for this rectangular prism is 76 square inches.
  ::答案是这个矩形棱柱的表面面积是76平方英寸。

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  Review
  ::回顾

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  Use the formula for surface area to find the surface area of each rectangular prism.
  ::使用表面积公式查找每个矩形棱柱的表面积。

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  1. A rectangular prism with a length of 10 in, width of 8 in and height of 5 inches.
     ::长10英寸长的矩形棱镜,宽8英寸,高度5英寸。
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  2. A rectangular prism with a length of 8 in, width of 8 in and height of 7 inches.
     ::长8英寸长的矩形棱镜,宽8英寸长,高7英寸长。
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  3. A rectangular prism with a length of 12 m, width of 4 m and height of 6 meters.
     ::长12米长的矩形棱镜,宽4米,高度6米。
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  4. A rectangular prism with a length of 10 in, a width of 6 in and a height of 7 inches.
     ::长10英寸长的矩形棱柱,宽6英寸高7英寸。
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  5. A rectangular prism with a length of 12 m, a width of 8 m and a height of 5 meters.
     ::长12米长的矩形棱柱,宽8米,高度5米。
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  6. A rectangular prism with a length of 9 ft, a width of 7 feet and a height of 6 feet.
     ::长9英尺长的长方形棱柱,宽7英尺,高6英尺。
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  7. A rectangular prism with a length of 10 m, a width of 8 m and a height of 2 m.
     ::长10米、宽8米、高2米的矩形棱柱。
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  8. A rectangular prism with a length of 6 ft, a width of 5 feet and a height of 3 feet.
     ::长6英尺长的矩形棱柱,宽5英尺,高度3英尺。
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  9. A rectangular prism with a length of 3 feet, a width of 6 feet and a height of 2 feet.
     ::长3英尺长的长方形棱柱,宽6英尺,高2英尺。
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  10. A rectangular prism with a length of 4 feet, a width of 4 feet and a height of 4 feet.
      ::长4英尺长的长方形棱柱,宽4英尺,高4英尺。
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  11. A rectangular prism with a length of 12 feet, a width of 9 feet and a height of 7 feet.
      ::长12英尺长的长方形棱镜,宽9英尺,高7英尺。
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  12. A rectangular prism with a length of 14 feet, a width of 11 feet and a height of 10 feet.
      ::长14英尺长的长方形棱镜,宽11英尺,高10英尺。
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  13. A rectangular prism with a length of 18 feet, a width of 16 feet and a height of 12 feet.
      ::长18英尺长的长方形棱镜,宽16英尺,高12英尺。
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  14. A rectangular prism with a length of 22 meters, a width of 18 meters and a height of 10 meters.
      ::长22米长的矩形棱镜,宽18米,高度10米。
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  15. A rectangular prism with a length of 21 meters, a width of 18 meters and a height of 17 meters.
      ::长21米长的矩形棱镜,宽18米,高度17米。

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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