节: 金字塔体积 | 8.12 金字塔数量

章节大纲

Brianna bought a unique candle as a gift for her friend’s birthday. The candle has a square base with a length of $\begin{align*}12 \ cm\end{align*}$ . The height of the candle is $\begin{align*}24 \ cm\end{align*}$ . The package says that the candle burns one hour for every 20 cubic centimeters of wax. How many hours will it take for the entire candle to burn?
::Brianna买了一支独特的蜡烛,作为她朋友生日的礼物。蜡烛有一个12厘米长的平方基。蜡烛的高度是24厘米。包裹上说蜡烛每20立方厘米蜡就烧一个小时。整个蜡烛要烧多久?

In this concept, you will learn to calculate the volume of pyramids .
::在这个概念中,你会学会计算金字塔的数量。

Volume
::量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量

is the measure of how much space a three-dimensional figure takes up or holds. Imagine a funnel. The funnel’s size will determine how much water the funnel will hold. If you fill it with water, the amount of water tells the volume of the funnel. Volume is often what you think of when you talk about measuring liquid or liquid capacity. Note that you measure volume in cubic units .
::度量一个三维图的大小。想象一个漏斗。漏斗的大小将决定漏斗的储水量。如果您用水填充漏斗, 水量会显示漏斗的体积。当您谈论测量液体或液体容量时, 量通常会想到量。注意用立方体单位量体积。

Pyramids are unusual because they are so much smaller at the top than they are at their base. The important thing to remember is that measuring volume involves filling up a . A pyramid has exactly one-third the volume of a cube.
::金字塔是不寻常的,因为它们在顶部比在底部要小得多。重要的是要记住测量体积需要填充一个。金字塔的体积正好是立方体的三分之一。

The formula for finding the volume of pyramids is:
::寻找金字塔体积的公式是:

$\begin{align*}V = \frac{1}{3} Bh\end{align*}$

$\begin{align*}B\end{align*}$ represents the base of the pyramid and $\begin{align*}h\end{align*}$ is the height of the pyramid. Pyramids can have bases of any shape. Pyramids can have triangular, rectangular, or square bases. That means you need to choose the appropriate formula for finding the area of the base, or $\begin{align*}B\end{align*}$ .
::V=13Bh B 代表金字塔的底部, h 是金字塔的高度。金字塔可以有任何形状的底部。金字塔可以有三角、矩形或平方基。这意味着您需要选择合适的公式来寻找基座区域, 或 B 。

Here are some common area formulas:
::以下是一些共同区域公式:

$\begin{align*}\text{Rectangle}: A = lw\end{align*}$
::矩形: A=lw

$\begin{align*}\text{Square}: A = s^2\end{align*}$
::平方: A=s2

$\begin{align*}\text{Triangle}: A = \frac{1}{2} bh\end{align*}$
::三角形:A=12bh

When given a pyramid, the first thing you need to do is determine the shape of the base. Then you will know which formula to use to find the base area. Once you have the base area, you put it into the volume formula along with the height of the pyramid and then solve for $\begin{align*}V\end{align*}$ .
::当给定一个金字塔时,你首先需要做的是确定基数的形状。然后,你会知道要使用哪种公式来寻找基区。一旦有了基区,你就会将其与金字塔的高度一起放入音量公式中,然后解决 V 。

Let’s look at an example.
::让我们举个例子。

What is the volume of the pyramid below?
::下面金字塔的体积是多少?

First, figure out the shape of the base. There are two pairs of parallel sides that meet at right angles , so it must be a rectangle . You need to use the area formula for to find $\begin{align*}B\end{align*}$ , the base area.
::首先, 找出基点的形状。有两对平行的两边在右角度相交, 所以它必须是一个矩形。您需要使用区域公式来找到 B, 即基点区域。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (11) (6.3) \\ A & = & 69.3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& lw \\ A &=& (11)(6.3) \\ A &=& 69.3 \end{array}\end{align*}$
::A=lwA=(11)(6.3)A=69.3

Next, substitute the base area, $\begin{align*}A\end{align*}$ , and the height into the formula for the volume.
::其次,将基准区域A和高度替换为音量公式的高度。

$\begin{aligned} \begin{array}{rcl} V & = & \frac{1}{3} B h \\ V & = & \frac{1}{3} (69.3) (15) \\ V & = & 346.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (69.3)(15) \\ V &=& 346.5 \end{array}\end{align*}$

The answer is 346.5.
::V=13BhV=13(69.3)(15)V=346.5。答案是346.5。

The volume of the pyramid is $\begin{align*}346.5 \ in^3\end{align*}$ .
::金字塔的体积是346.5英寸3。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Brianna’s melting gift.
::Brianna的融化礼物问题。

Brianna bought a candle with a square base length of $\begin{align*}12 \ cm\end{align*}$ and a height of $\begin{align*}24 \ cm\end{align*}$ . She wants to know how long the candle will burn at a rate of $\begin{align*}20 \ cm^3\end{align*}$ per hour.
::Brianna买了一支蜡烛,其基本长度为12厘米,高度为24厘米。她想知道蜡烛会以每小时20厘米的速度燃烧多久。

First, use the area formula for squares to find $\begin{align*}B\end{align*}$ , the base area.
::首先,使用方形的区域公式来找到B,即基区。

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & (12)^{2} \\ A & = & 144 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (12)^2 \\ A &=& 144 \end{array}\end{align*}$
::A=S2A=(12)2A=144

Next, substitute the base area, $\begin{align*}A\end{align*}$ , and the height into the formula for the volume.
::其次,将基准区域A和高度替换为音量公式的高度。

$\begin{aligned} \begin{array}{rcl} V & = & \frac{1}{3} B h \\ V & = & \frac{1}{3} (144) (24) \\ V & = & 1152 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (144)(24) \\ V &=& 1152 \end{array}\end{align*}$
::V=13BhV=13(144)(24)(24)V=1152

Then, find the number of hours the candle will burn.
::然后找出蜡烛会燃烧的小时数

$\begin{aligned} \begin{array}{rcl} # hours & = & \frac{1152}{20} \\ # hours & = & 57.6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \# \ \text{hours} &=& \frac{1152}{20} \\ \# \ \text{hours} &=& 57.6 \end{array}\end{align*}$
::#小时=115220#小时=57.6

The answer is 57.6.
::答案是57.6

The candle will burn 57.6 hours.
::蜡烛将燃烧57.6小时。

Example 2
::例2

A triangular pyramid has a volume of $\begin{align*}266 \ \text{cubic feet}\end{align*}$ and a base area of $\begin{align*}42 \ \text{square feet}\end{align*}$ . What is its height?
::三角金字塔的体积为266立方英尺,基底面积为42平方英尺。

First, substitute what you know into the formula for the volume of the pyramid.
::首先,用你所知道的来代替金字塔的体积

$\begin{aligned} \begin{array}{rcl} V & = & \frac{1}{3} B h \\ 266 & = & \frac{1}{3} (42) h \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ 266 &=& \frac{1}{3}(42)h \end{array}\end{align*}$
::V=13Bh266=13(42)h

Next, use algebra to solve for the height, $\begin{align*}h\end{align*}$ .
::下一个,使用代数解析高度,h。

$\begin{aligned} \begin{array}{rcl} 266 & = & \frac{1}{3} (42) h \\ 266 & = & 14 h \\ \frac{266}{14} & = & \frac{14 h}{14} \\ h & = & 19 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 266 &=& \frac{1}{3}(42)h \\ 266 &=& 14h \\ \frac{266}{14} &=& \frac{14h}{14} \\ h &=& 19 \end{array}\end{align*}$
::266=13(42)h266=14h26614=14h14h14h=19

The answer is 19.
::答案是19个

The height of the pyramid is $\begin{align*}19 ft\end{align*}$ .
::金字塔的高度是19英尺

Example 3
::例3

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

First, use the area formula for squares to find $\begin{align*}B\end{align*}$ , the base area.
::首先,使用方形的区域公式来找到B,即基区。

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & (5.5)^{2} \\ A & = & 30.25 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (5.5)^2 \\ A &=& 30.25 \end{array}\end{align*}$
::A=S2A=(5.5)2A=30.25

Next, substitute the base area, $\begin{align*} A\end{align*}$ , and the height into the formula for the volume.
::其次,将基准区域A和高度替换为音量公式的高度。

$\begin{aligned} \begin{array}{rcl} V & = & \frac{1}{3} B h \\ V & = & \frac{1}{3} (30.25) (4) \\ V & = & 40.33 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (30.25)(4) \\ V &=& 40.33 \end{array}\end{align*}$
::V=13BhV=13(30.25)(4)V=40.33

The answer is 40.33.
::答案是40.33。

The volume of the pyramid is $\begin{align*}40.33 \ in^3\end{align*}$ .
::金字塔的体积是40.33英寸

Example 4
::例4

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

First, use the area formula for squares to find $\begin{align*}B\end{align*}$ , the base area.
::首先,使用方形的区域公式来找到B,即基区。

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & (8) 2 \\ A & = & 64 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (8)2 \\ A &=& 64 \end{array}\end{align*}$
::A=S2A=(8)2A=64

Next, substitute the base area, $\begin{align*}A\end{align*}$ , and the height into the formula for the volume.
::其次,将基准区域A和高度替换为音量公式的高度。

$\begin{aligned} \begin{array}{rcl} V & = & \frac{1}{3} B h \\ V & = & \frac{1}{3} (64) (6) \\ V & = & 128 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (64)(6) \\ V &=& 128 \end{array}\end{align*}$
::V=13BhV=13(64)(6)V=128

The answer is 128.
::答案是128个

The volume of the pyramid is $\begin{align*}128 \ cm^3\end{align*}$ .
::金字塔的体积是128厘米3。

Example 5
::例5

Find the volume of a rectangular pyramid with a length of $\begin{align*}10 \ cm\end{align*}$ , a width of $\begin{align*}8 \ cm\end{align*}$ and a height of $\begin{align*}9 \ cm\end{align*}$ .
::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。

First, use the area formula for rectangles to find $\begin{align*}B\end{align*}$ , the base area.
::首先,使用矩形的区域公式来找到B,即基区。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (10) (8) \\ A & = & 80 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& lw \\ A &=& (10)(8) \\ A &=& 80 \end{array}\end{align*}$
::A=lwA=(10)(10)(8)A=80

Next, substitute the base area, $\begin{align*}A\end{align*}$ , and the height into the formula for the volume.
::其次,将基准区域A和高度替换为音量公式的高度。

$\begin{aligned} \begin{array}{rcl} V & = & \frac{1}{3} B h \\ V & = & \frac{1}{3} (80) (9) \\ V & = & 240 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (80)(9) \\ V &=& 240 \end{array}\end{align*}$
::V=13BhV=13(80)(9)V=240

The answer is 240.
::答案是240

The volume of the pyramid is $\begin{align*}240 \ cm^3\end{align*}$ .
::金字塔的体积是240立方厘米。

Review
::回顾

Answer each of the following questions.
::回答下列问题中的每一个问题。

1. What is the formula that you would need for finding the volume of a pyramid?
::1. 要找到金字塔的体积,你需要什么公式?

2. When you see a capital B in a formula it means that you are looking for the perimeter or area of the base?
::2. 在公式中看到首都B时,表示正在寻找基地的周边或区域?

3. What is the name of the figure pictured below?
::3. 以下图示的数字叫什么?

4. What is the shape of the base?
::4. 基地的形状是什么?

5. What is the volume of the figure?
::5. 数字的数量是多少?

6. What is the name of the figure pictured above?
::6. 上述数字的名称是什么?

7. What is the difference between this figure and the last figure?
::7. 这一数字与上一个数字之间有什么区别?

8. What is the volume of this figure?
::8. 这个数字的数量是多少?

9. What is the shape of the base?
::9. 基地的形状是什么?

Use what you have learned to solve each of the following problems. Round your answer to the nearest hundredths place.
::使用您所学的知识来解决以下每一个问题。将您的答案四舍五入到最近的百个地方。

10. A square pyramid has a base with sides of 6 yards each and a volume of 175 cubic yards. What is its height?
::10. 方形金字塔的基座各有6码,体积175立方码。

11. Claire has a perfume bottle shaped like a triangular pyramid. Its base area is 48 square centimeters, and its height is 28 centimeters. How much does the bottle hold when it is exactly half full?
::11. 克莱尔的香水瓶形状像三角金字塔,其底部面积为48平方厘米,高度为28厘米。

12. Find the volume of a square pyramid with a base of 4 inches and a height of 6 inches.
::12. 找到一个基数为4英寸、高度为6英寸的平方金字塔的体积。

13. Find the volume of a rectangular pyramid with a length of 5 inches, a width of 7 inches and a height of 8 inches.
::13. 查找长5英寸、宽7英寸、高8英寸长的长方形金字塔的体积。

14. Find the volume of a square pyramid with a base of 8 meters and a height of 12 meters.
::14. 找到一个基数为8米、高度为12米的平方金字塔的体积。

15. Find the volume of a square pyramid with a base of 13.5 meters and a height of 15 meters.
::15. 找到一个基数为13.5米、高度为15米的平方金字塔的体积。

Review (Answers)
::回顾(答复)

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。

Resources
::资源

章节大纲

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源