10.14 金字塔数量

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  金字塔体积

  

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  Michael and Juanita are interested in the amount of sand required to build a sand pyramid of the size that they see while at a sand sculpture exhibit.  They carefully measure the width of the square base of the sand pyramid which is 4 feet and the height of the pyramid which is 5 feet.  How much sand is required to build this sand pyramid?
  ::Michael Juanita 和 Michael Juanita 都对在沙雕展上建造他们所看到的大小的沙金字塔所需的沙子数量感兴趣。 他们仔细测量了沙金字塔的平方底部宽度,即4英尺,金字塔的高度为5英尺。 建造这个沙金字塔需要多少沙子?

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  In this concept, you will learn how to calculate the of a pyramid.
  ::在这个概念中,你会学会如何计算金字塔。

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  Finding the Volume of a Pyramid
  ::查找金字塔的音量

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  Pyramids are solid shapes that exist in three-dimensional space . A pyramid has sides that are triangular faces and a base that can be any shape.
  ::金字塔是三维空间中存在的固态。 金字塔有三角面的两面,是任何形状的基座。

  

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  Volume is the measure of how much space a three-dimensional figure takes up or holds. Volume is measured in cubic units , because you are dealing with three dimensions : length, width, and height.
  ::音量是衡量一个三维图的接收或持有空间大小的尺度。 音量用立方单位计量, 因为您处理的是三个维度: 长度、 宽度和高度。

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  You can start to understand the volume of a pyramid by comparing a pyramid with a figure that is similar to it. You know that a prism has length, width and height. The parallel bases of a prism can be any polygon .  This one has a square base; so it is called a cube.
  ::您可以通过比较金字塔和类似数字来开始理解金字塔的体积。 您知道棱晶有长度、 宽度和高度。 棱晶的平行基数可以是任何多边形。 此基数有一个平方基; 因此它被称为立方体 。

  

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  Now think about a pyramid. It has a base that can be any polygon.  The base of this pyramid is a square.
  ::现在想想金字塔。 它有一个可以成为任何多边形的基数。 这个金字塔的基数是方形 。

  

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  These are both similar given that they have square bases. Think about the volume of a pyramid by looking at the volume of the cube or prism. Look at this picture that demonstrates the relationship of a cube to a square pyramid:
  ::两者都相似, 因为它们有平方基。 通过查看立方体或棱柱的体积来思考金字塔的体积。 看看这个显示立方体与平方金字塔关系的图片 :

  

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  To find the volume of this cube, multiply the base times the width times the height. To find the volume of the pyramid, take the area of the base, $\begin{array}{r}B\end{array}$ and multiply it times the height and then multiply it by $\begin{array}{r}\frac{1}{3}\end{array}$ . Here is the formula for finding the volume of a pyramid.
  ::要找到此立方体的体积, 请乘以宽度乘以宽度乘以高度。 要找到金字塔的体积, 请取基数区域, B 乘以基数区域, 然后乘以高度, 然后乘以13 。 这是用于查找金字塔体积的公式 。

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  $\begin{array}{r}V=\frac{1}{3}Bh\end{array}$
  ::V=13Bh

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  One thing to keep in mind is that pyramids can be tricky because they can have many different bases.   $\begin{array}{r}B\end{array}$  in the formula means that you need to find the area of the base. If the base is a square, you will need to use the formula for area of a square, $\begin{array}{rl}B& =s^2\end{array}$ .  If the base is a rectangle , the area of the base is $\begin{array}{rl}B& =lw\end{array}$ . If the base is a triangle , use the formula $\begin{array}{rl}B& =\frac{bh}{2}\end{array}$ . Making sure that you have the correct formula is essential in your work with pyramids.
  ::需要记住的一件事是,金字塔可能很棘手,因为它们可以有许多不同的基点。公式中 B 表示您需要找到基点的区域。如果基点是一个正方形,您需要使用方块的公式,B=s2。如果基点是一个矩形,则基点的区域是B=lw。如果基点是一个三角,请使用公式B=bh2。确保您拥有正确的公式对于您处理金字塔的工作至关重要。

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  Let's look at an example. What is the volume of the pyramid below?
  ::举个例子,下面金字塔的体积是多少?

  

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  First, decide what shape the base of the pyramid is. One side is 10 centimeters and the other is 6 centimeters, so it must be a rectangle. Use the area formula for to find $\begin{array}{r}B\end{array}$ , the base area.
  ::首先, 决定金字塔底部的形状 。 一边是 10 厘米, 另一边是 6 厘米, 所以它必须是一个矩形 。 使用区域公式找到 B , 即基区 。

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  $$\begin{array}{rl}B& =lw\\ B& =10(6)\\ B& =60c{m}^2\end{array}$$

  
  ::B=lwB=10(6)B=60平方厘米2

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  The area of this pyramid’s base is 60 square centimeters. Now multiply this by the height and $\begin{array}{r}\frac{1}{3}\end{array}$   according to the formula.
  ::此金字塔基座的面积是60平方厘米。 现在根据公式乘以高度, 乘以13 。

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  $$\begin{array}{rl}V& =\frac{1}{3}Bh\\ V& =\frac{1}{3}(60)(15)\\ V& =\frac{900}{3}\\ V& =300c{m}^3\end{array}$$

  
  ::V=13BhV=13(60)(15)V=9003V=300cm3

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  The answer is the volume of the pyramid is 300 cubic centimeters.
  ::答案是金字塔的体积是300立方厘米

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  Remember, you measure volume in three dimensions, so write the answer in cubic units.
  ::记住,你用三个维度来测量体积, 所以用立方单位来写答案 。

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  Let's look at another example.
  ::让我们再举一个例子。

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  Find the volume of the figure below.
  ::查找下图的音量。

  

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  This time the base is a triangle, so you'll need to use the area formula for triangles to find the base area. Be careful not to confuse the height of the base triangle with the height of the whole pyramid.
  ::当基点是一个三角形时, 您需要使用三角形的区域公式来找到基点区域。 注意不要将基点三角形的高度与整个金字塔的高度混为一谈 。

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  $$\begin{array}{rl}B& =\frac{1}{2}bh\\ B& =\frac{1}{2}(8)(3)\\ B& =\frac{24}{2}\\ B& =12in{.}^2\end{array}$$

  
  ::B=12bhB=12(8)(3)B=242B=12英寸。

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  The base area for this triangular pyramid is 12 square inches. Put this into the formula and solve for $\begin{array}{r}V\end{array}$ , volume.
  ::此三角金字塔的基本区域是 12 平方英寸。 将其输入公式并解析 V, 音量 。

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  $$\begin{array}{rl}V& =\frac{1}{3}Bh\\ V& =\frac{1}{3}(12)(17)\\ V& =\frac{204}{3}\\ V& =68in{.}^3\end{array}$$

  
  ::V=13BhV=13(12)(17V=2043V=68),3英寸

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  The answer is the volume of this pyramid is 68 cubic inches.
  ::答案是金字塔的体积是68立方英寸

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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 Michael and Juanita's interest in the sand sculpture pyramid.
  ::早些时候,你被问及 迈克尔和胡安妮塔对沙雕金字塔的兴趣

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  They want to figure out the volume of sand required to create a sand sculpture pyramid with base 4 feet and height 5 feet:
  ::他们想弄清楚沙子的体积, 以建立沙雕金字塔, 底座是4英尺,高度是5英尺:

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  First, determine the shape of the base and use the correct formula to find the base area.  Since the base is a square, use the area formula for squares to find the base area.
  ::首先, 确定基点的形状, 并使用正确的公式来查找基点区域。 由于基点是一个平方, 使用方块的区域公式来查找基点区域 。

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  $$\begin{array}{rl}B& =s^2\\ B& =(4{)}^2\\ B& =16f{t}^2\end{array}$$

  The base area of the pyramid is 16 square centimeters.
  ::B=2B=(4)2B=16ft2 金字塔的基本面积为16平方厘米。

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  Next, plug the area of the base, and the height of the pyramid into the formula for volume of a pyramid and multiply the values together.
  ::接下去,将基底的面积和金字塔的高度插入金字塔体积的公式中,并将数值相乘。

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  $$\begin{array}{rl}V& =\frac{1}{3}Bh\\ V& =\frac{1}{3}(16)(5)\end{array}$$

  
  ::V=13BhV=13(16)(5)

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  $$\begin{array}{rl}V& =\frac{80}{3}\end{array}$$

  
  ::V=803

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  Then, divide by 3 for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将答案除以3,确保包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{80}{3}\\ V& =26.67f{t}^3\end{array}$$

  
  ::V=803V=26.67平方英尺

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  The answer is the volume of sand required to create the sand sculpture pyramid is 26.67 cubic feet.
  ::答案是创造沙雕金字塔所需的沙积是26.67立方英尺。

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  Example 2
  ::例2

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  Felice bought the candle below for her friend’s birthday. The package says that the candle burns one hour for every 20 cubic centimeters of wax. What is the volume of the candle so that Felice can figure out how long it will burn?
  ::Felice为她朋友的生日买了下面的蜡烛。 包裹上说蜡烛每20立方厘米的蜡就燃烧一小时。 蜡烛的体积是多少才能让Felice知道它会燃烧多久?

  

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  First, determine the shape of the base and use the correct formula to find the base area.   Since the base is a square, you use the area formula for squares to find the base area.
  ::首先,确定基点的形状,并使用正确的公式来查找基点区域。由于基点是一个正方形,所以使用方块的区域公式来查找基点区域。

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  $$\begin{array}{rl}B& =s^2\\ B& =(12{)}^2\\ B& =144c{m}^2\end{array}$$

  
  ::B=S2B=(12)2B=144厘米2

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  The base area of the pyramid is 144 square centimeters.
  ::金字塔的基本面积为144平方厘米。

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  Next, plug the area of the base, and the height of the pyramid into the formula for volume of a pyramid and multiply the values together.
  ::接下去,将基底的面积和金字塔的高度插入金字塔体积的公式中,并将数值相乘。

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  $$\begin{array}{rl}V& =\frac{1}{3}Bh\\ V& =\frac{1}{3}(144)(24)\end{array}$$

  
  ::V=13BhV=13(144)(24)

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  $$\begin{array}{rl}V& =\frac{3456}{3}\end{array}$$

     
  ::V=34563

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  Then, divide by 3 for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将答案除以3,确保包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{3456}{3}\\ V& =1,152c{m}^3\end{array}$$

  
  ::V=34563V=1,152立方厘米

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  The answer is the volume of the candle is 1,152 cubic cm.
  ::答案是蜡烛的体积是1,152立方厘米。

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  Example 3
  ::例3

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  Find the volume of a square pyramid with a base of 8 cm and a height of 6 cm.
  ::查找方形金字塔的体积,其基数为8厘米,高度为6厘米。

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  First, determine the shape of the base and use the correct formula to find the base area.  The base is a square, so use the area formula for squares to find the base area.

  $$\begin{array}{rl}B& =s^2\\ B& =(8{)}^2\\ B& =64c{m}^2\end{array}$$

  
  ::首先, 确定基点的形状, 并使用正确的公式查找基区 。 基点是一个平方, 所以使用方的面积公式查找基区 。 B=s2B=( 8) 2B=64 cm2

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  The base area of the pyramid is 64 square centimeters.
  ::金字塔的基本面积为64平方厘米。

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  Next, plug the area of the base, and the height of the pyramid into the formula for volume of a pyramid and multiply the values together.
  ::接下去,将基底的面积和金字塔的高度插入金字塔体积的公式中,并将数值相乘。

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  $$\begin{array}{rl}V& =\frac{1}{3}Bh\\ V& =\frac{1}{3}(64)(6)\end{array}$$

  
  ::V=13BhV=13(64)(6)

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  $$\begin{array}{rl}V& =\frac{384}{3}\end{array}$$

  
  ::V=3843

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   Then, divide by 3 for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将答案除以3,确保包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{384}{3}\\ V& =128{cm}^3\end{array}$$

  
  ::V=3843V=128厘米3

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  The answer is the volume of the pyramid is 128 cubic cm.
  ::答案是金字塔的体积是128立方厘米

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  Example 4
  ::例4

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  Find the volume of a rectangular pyramid with a length of 10 cm, a width of 8 cm and a height of 9 cm.
  ::查找长10厘米、宽8厘米、高度9厘米的长方形金字塔的体积。

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  First, determine the shape of the base and use the correct formula to find the base area.  Since the base is a rectangle, use the area formula for squares to find the base area.
  ::首先, 确定基点的形状, 并使用正确的公式来查找基点区域。 由于基点是一个矩形, 使用方形的区域公式来查找基点区域 。

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  $$\begin{array}{rl}B& =lw\\ B& =(10)(8)\\ B& =80c{m}^2\end{array}$$

  The base area of the pyramid is 80 square centimeters.
  ::B=lwB=(10)(8)B=80cm2 金字塔的基本面积为80平方厘米。

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  Next, plug the area of the base, and the height of the pyramid into the formula for volume of a pyramid and multiply the values together.

  $$\begin{array}{rl}V& =\frac{1}{3}Bh\\ V& =\frac{1}{3}(80)(9)\end{array}$$

  
  

  $$\begin{array}{rl}V& =\frac{720}{3}\end{array}$$

  
  ::下一步,将基底面积和金字塔高度插入金字塔体积的公式中,并将数值乘以13BhV=13(80)(9)V=7203

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  Then, divide by 3 for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将答案除以3,确保包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{720}{3}\\ V& =240c{m}^3\end{array}$$

  
  ::V=72003V=240立方米

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  The answer is the volume of the pyramid is 240 cubic cm.
  ::答案是金字塔的体积是240立方厘米

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  Example 5
  ::例5

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  Find the volume of a square pyramid with a base of 5.5 in and a height of 4 in.
  ::找到一个方形金字塔的体积,基数5.5英寸,高4英寸。

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  First, determine the shape of the base and use the correct formula to find the base area.  The base is a square, so use the area formula for squares to find the base area.
  ::首先, 确定基点的形状, 并使用正确的公式来查找基点区域。 基点是一个平方, 所以使用方块的面积公式来查找基点区域 。

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  $$\begin{array}{rl}B& =s^2\\ B& =(5.5{)}^2\\ B& =30.25i{n}^2\end{array}$$

  
  ::B=S2B=(5.5)2B=30.25英寸2

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  The base area of the pyramid is 30.25 square centimeters.  
  ::金字塔的基本面积为30.25平方厘米。

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  Next, plug the area of the base, and the height of the pyramid into the formula for volume of a pyramid and multiply the values together.
  ::接下去,将基底的面积和金字塔的高度插入金字塔体积的公式中,并将数值相乘。

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  $$\begin{array}{rl}V& =\frac{1}{3}Bh\\ V& =\frac{1}{3}(30.25)(4)\end{array}$$

  
  ::V=13BhV=13(30.25)(4)

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  $$\begin{array}{rl}V& =\frac{121}{3}\end{array}$$

  
  ::V=1213

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  Then, divide by 3 for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将答案除以3,确保包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{121}{3}\\ V& =40.33i{n}^3\end{array}$$

  
  ::V=1213V=40.33英寸3

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  The answer is the volume of the pyramid is 40.33 cubic inches.
  ::答案是金字塔的体积是40.33立方英寸

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  Review
  ::回顾

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  Find the volume of each of the following pyramids.
  ::查找以下金字塔的体积 。

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  1. A square pyramid with a base of 6 ft and a height of 9 ft.
     ::平方金字塔,基数6英尺,高9英尺
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  2. A square pyramid with a base of 8 m and a height of 10 m.
     ::方形金字塔,基数8米,高度10米。
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  3. A square pyramid with a base of 11 in and a height of 13 in.
     ::方形金字塔,基数11英寸,高度13英寸。
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  4. A square pyramid with a base of 9 ft and a height of 14 ft.
     ::方形金字塔,基数9英尺,高14英尺
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  5. A square pyramid with a base of 4.5 in and a height of 5 inches.
     ::方形金字塔,基数4.5英寸,身高5英寸。
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  6. A rectangular pyramid with a base length of 4 in, a base width of 3 in and a height of 5 in.
     ::长方形金字塔,基长4英寸,基宽3英寸,高5英寸。
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  7. A rectangular pyramid with a base length of 5 ft, a base width of 4 ft and a height of 6 ft.
     ::长方形金字塔,基长5英尺,基宽4英尺,高6英尺。
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  8. A rectangular pyramid with a base length of 7 m, a base width of 4 m and a height of 9 m.
     ::长方形金字塔,基长7米,基宽4米,高9米。
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  9. A triangular pyramid with a base length of 5 in and a base height of 4 inches with a pyramid height of 6 inches.
     ::三角金字塔,基长5英寸,基高4英寸,金字塔高6英寸。
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  10. A triangular pyramid with a base length of 8 ft and a base height of 7 ft with a pyramid height of 9 ft.
      ::三角金字塔,基长8英尺,基高7英尺,金字塔高9英尺。
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  11. A square pyramid with a base of 8 feet and height of 4 feet.
      ::方形金字塔 基数8英尺高4英尺
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  12. A rectangular pyramid with a length of 5 inches, a width of 4 inches and a height of 6 inches.
      ::长5英寸长的矩形金字塔,宽4英寸,高6英寸。
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  13. A square pyramid with a base of 3.5 feet and a height of 6.5 feet.
      ::方形金字塔,基数3.5英尺,高6.5英尺。
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  14. A square pyramid with a base of 6.5 feet and a height of 8.5 feet.
      ::方形金字塔,基数6.5英尺,高8.5英尺。
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  15. A rectangular pyramid with a width of 4 feet, a length of 6 feet and a height of 7.5 feet.
      ::一个长方形金字塔,宽4英尺,长6英尺,高7.5英尺。

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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 。