Course: 10.11 三角棱晶体量

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三角棱晶体体积
Maggie is enjoying her favorite chocolate. She wonders how much chocolate can fit into the packaging, which is in the shape of a triangular prism . She measures the cardboard prism and finds that the triangle end, or base, is 1.4 inches, and the height of the triangle is 1.25 inches. The height of the bar when stood on its end is 8.25 inches. What is the of the chocolate package?
::Maggie正在享受她最喜欢的巧克力。她想知道有多少巧克力可以融入包装中, 包装是三角棱柱形的。她测量了纸板棱柱, 发现三角形的底部是1.4英寸, 三角形的高度是1. 25英寸。站在底部的酒吧高度是8. 25英寸。什么是巧克力包件?

In this concept, you will learn how to calculate the volume of a triangular prism.
::在这个概念中,你会学会如何计算三角棱镜的体积。

Finding the Volume of a Triangular Prism
::查找三角棱柱的量

With a triangular prism, the two parallel faces are triangles and the other faces are .
::在三角棱镜中,两个平行面孔是三角形,其他面孔是三角形。

You calculate the volume of triangular prisms almost the same way that you find the . You still use the formula $\begin{align*}V = Bh\end{align*}$ . However, this time the base of the prism is a triangle, not a rectangle . Therefore, you need to use the area formula for a triangle to find the area of the base, $\begin{align*}B\end{align*}$ . Then you can multiply this amount by the height of the rectangle to find the volume of the triangular prism.
::您计算三角棱晶体积的方式几乎与您找到三角棱晶体的方式相同。您仍然使用公式V=Bh。但是, 这次棱晶的基数是三角形, 不是矩形。因此, 您需要使用三角形的面积公式来找到基数区域, B。然后您可以将这个数值乘以矩形的高度, 以找到三角棱晶体的体积。

Let’s look at an example.
::让我们举个例子。

What is the volume of this triangular prism where the height of the triangular base is 7 cm and the width of the base of the triangle is 5 cm?
::三角基点的高度是 7 cm 三角基点的宽度是 5 cm 。

First ,you need to find the area of the triangular base. You use the formula for the area of a triangle which is $\begin{align*}\frac{1}{2} bh\end{align*}$ . Remember, you use the height and base measurements for the triangular face , not the height, $\begin{align*}H\end{align*}$ , measurement for the whole prism which is the length of the rectangle.
::首先,您需要找到三角基点的面积。您需要为三角形的面积使用公式,即 12bh。记住,您要对三角形的面部使用高度和基度测量,而不是高度, H,对矩形长度的整个棱镜的测量。

So there are two things that you need to accomplish: You need to find the area of one of the triangular bases, and then you can take that measurement and multiply it with the height of the entire prism.
::所以,有两件事你需要完成: 你需要找到三角基点之一的区域, 然后你就可以用这个测量法乘以整个棱镜的高度。

$\begin{aligned} V & = B H \\ B & = \frac{1}{2} b h \\ B & = \frac{1}{2} (5) (7) \\ B & = 35 \\ V & = (35) H \\ V & = (35) (12) \\ V & = 420 c m^{3} \end{aligned}$
$\begin{align*}V & = BH\\ B & = \frac{1}{2} bh\\ B & = \frac{1}{2}(5)(7)\\ B & = 35\\ V & = (35)H\\ V & = (35)(12)\\ V & = 420 \ cm^3\end{align*}$
::V=BHB=12bhB=12(5)(7)B=35V=(35)HV=(35)35(12)V=420立方厘米3

The answer is the volume of this triangular prism is $\begin{align*}420 \ centimeters^3\end{align*}$ .
::答案是三角棱镜的体积是420厘米3。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about how Maggie wants to find the volume of chocolate that will fit inside the packaging.
::之前,你被问到一个问题玛姬想如何找到巧克力的体积将适合包装。

The triangular base is 1.4 inches; the height of the triangle is 1.25 inches, and the height of the bar when stood on its end is 8.25 inches.
::三角基数为1.4英寸;三角形的高度为1.25英寸,顶端的横杆的高度为8.25英寸。

First, plug in the values for the area formula of a triangle to find the base area, $\begin{align*}B\end{align*}$ .
$\begin{aligned} B & = \frac{1}{2} b h \\ B & = \frac{1}{2} (1.4) (1.25) \\ B & = 0.7 (1.25) \\ B & = 0.88 i n^{2} \end{aligned}$
$\begin{align*}B & = \frac{1}{2} bh\\ B & = \frac{1}{2} (1.4) (1.25)\\ B & = 0.7 (1.25)\\ B & = 0.88 \ in^2 \end{align*}$
::首先,插入三角形区域公式的值以查找基区域,B.B=12bhbhB=12(1.4)(1.25)B=0.7(1.25)B=0.88

Next, plug the value, $\begin{align*}B\end{align*}$ , and the value for the height of the prism, $\begin{align*}H\end{align*}$ , into the volume formula and multiply the values together.
::下一步,将数值 B 和棱晶高度的数值 H 插入音量公式,并将数值相乘。

$\begin{aligned} V & = B H \\ V & = 0.88 \times 8.25 \end{aligned}$
$\begin{align*}V & = BH\\ V & = 0.88 \times 8.25\\\end{align*}$
::V=BHV=0.88x8.25

$\begin{aligned} V & = 0.88 \times 8.25 \\ V & = 7.26 \end{aligned}$
$\begin{align*}V&= 0.88\ \times\ 8.25\\ V&= 7.26\end{align*}$
::V=0.88 × 8.25V= 7.26

Then, record the answer, making sure to include the appropriate unit of measurement.
::然后记录答案,确保包括适当的计量单位。

$\begin{aligned} V = 7.26 {i n}^{3} \end{aligned}$
$\begin{align*}V = 7.26\ {in}^{3}\end{align*}$
::V=7.26英寸3 V=7.26

The answer is the package has a volume of 7.26 cubic inches.
::答案是包件的体积为7.26立方英寸

Example 2
::例2

Every year Jeanie gets a bottle of her favorite perfume for her birthday. The perfume comes in a bottle shaped like a triangular prism. How much perfume does the bottle hold when it is full?
::珍妮每年为生日买一瓶她最喜欢的香水。香水是用一个装成三角棱晶的瓶子装成的。瓶子满满时有多少香水?

First, plug in the values for the area formula of a triangle to find the base area, $\begin{align*}B\end{align*}$ .
::首先,插入三角形区域公式的值,以找到基区域,B。

$\begin{aligned} B & = \frac{1}{2} b h \\ B & = \frac{1}{2} (6) (4) \\ B & = 3 (4) \\ B & = 12 c m^{2} \end{aligned}$
$\begin{align*}B & = \frac{1}{2} bh\\ B & = \frac{1}{2} (6) (4)\\ B & = 3 (4)\\ B & = 12 \ cm^2 \end{align*}$
::B=12bhB=12(6)(4)B=3(4)B=12cm2

Next, plug the triangle area value, $\begin{align*}B\end{align*}$ , and the value for the height of the prism, $\begin{align*}H\end{align*}$ , into the volume formula and multiply the values together.
::下一步,在音量公式中插入三角区域值,B,以及棱晶高度的数值,H,并同时乘以数值。

$\begin{aligned} V & = B H \\ V & = 12 \times 9 \end{aligned}$
$\begin{align*}V & = BH\\ V & = 12 \times 9\\\end{align*}$
::V=BHV=12x9

$\begin{aligned} V & = 12 \times 9 \\ V & = 108 \end{aligned}$
$\begin{align*}V&= 12\ \times\ 9\\ V&= 108\end{align*}$
::V=12 × 9V=108

Then, record the answer, making sure to include the appropriate unit of measurement.
::然后记录答案,确保包括适当的计量单位。

$\begin{aligned} V = 108 {c m}^{3} \end{aligned}$
$\begin{align*}V = 108\ {cm}^{3}\end{align*}$
::V=108厘米3

The answer is the perfume bottle holds 108 cubic centimeters of perfume when full.
::答案是香水瓶装满108立方厘米的香水

Example 3
::例3

What is the volume of a triangular prism with the following dimensions : $\begin{align*}b = 12 \ in, \ h = 10 \ in, \ H = 15 \ in\end{align*}$
::具有以下维度的三角边棱晶面的体积是多少 : b= 12 英寸, h= 10 英寸, H= 15 英寸

First, plug in the values for the area formula of a triangle to find the base area, $\begin{align*}B\end{align*}$ .
$\begin{aligned} B & = \frac{1}{2} b h \\ B & = \frac{1}{2} (12) (10) \\ B & = 6 (10) \\ B & = 60 i n^{2} \end{aligned}$
$\begin{align*}B & = \frac{1}{2} bh\\ B & = \frac{1}{2} (12) (10)\\ B & = 6 (10)\\ B & = 60 \ in^2 \end{align*}$
::首先,插入三角形区域公式的值以查找基区域,B.B=12bhB=12(12) (10)B=6(10)10B=60 in2

Next, plug the triangle area value, $\begin{align*}B\end{align*}$ , and the value for the height of the prism, $\begin{align*}H\end{align*}$ , into the volume formula and multiply the values together.
$\begin{aligned} V & = B H \\ V & = 60 \times 15 \end{aligned}$
$\begin{align*}V & = BH\\ V & = 60 \times 15\\\end{align*}$
::下一步,在音量公式中插入三角区域值、B和棱晶高度的数值H,并同时乘以数值。V=BHV=60x15

$\begin{aligned} V & = 60 \times 15 \\ V & = 900 \end{aligned}$
$\begin{align*}V&= 60\ \times\ 15\\ V&= 900\end{align*}$
::V=60 × 15V=900

Then, record the answer, making sure to include the appropriate unit of measurement.
::然后记录答案,确保包括适当的计量单位。

$\begin{aligned} V = 900 {i n}^{3} \end{aligned}$
$\begin{align*}V = 900\ {in}^{3}\end{align*}$
::V=900 英寸3

The answer is the volume of this triangular prism is $\begin{align*}900 \ in^3\end{align*}$ .
::答案是三角棱镜的体积是900英寸3

Example 4
::例4

What is the volume of a triangular prism with the following dimensions: $\begin{align*}b = 7 \ cm, \ h = 5 \ cm, \ H = 9 \ cm\end{align*}$
::具有以下维度的三角棱晶体体积是多少: b= 7 厘米, h= 5 厘米, H= 9 厘米

First, plug in the values for the area formula of a triangle to find the base area, $\begin{align*}B\end{align*}$ .
$\begin{aligned} B & = \frac{1}{2} b h \\ B & = \frac{1}{2} (7) (5) \\ B & = 3.5 (5) \\ B & = 17.5 c m^{2} \end{aligned}$
$\begin{align*}B & = \frac{1}{2} bh\\ B & = \frac{1}{2} (7) (5)\\ B & = 3.5 (5)\\ B & = 17.5 \ cm^2 \end{align*}$
::首先,插入三角形区域公式的值以找到基区域,B.B=12bhbhB=12(7)(7)(5)B=3.5(5)B=17.5cm2

Next, plug the triangle area value, $\begin{align*}B\end{align*}$ , and the value for the height of the prism, $\begin{align*}H\end{align*}$ , into the volume formula and multiply the values together.
$\begin{aligned} V & = B H \\ V & = 17.5 \times 9 \end{aligned}$
$\begin{align*}V & = BH\\ V & = 17.5 \times 9\\\end{align*}$
::下一步,在音量公式中插入三角区域值、B和棱晶高度的数值H,然后将数值相乘。 V=BHV=17.5x9

$\begin{aligned} V & = 17.5 \times 9 \\ V & = 157.5 \end{aligned}$
$\begin{align*}V&= 17.5\ \times\ 9\\ V&= 157.5\end{align*}$
::V=17.5×9V=157.5

Then, record the answer, making sure to include the appropriate unit of measurement.
::然后记录答案,确保包括适当的计量单位。

$\begin{aligned} V = 157.5 {c m}^{3} \end{aligned}$
$\begin{align*}V = 157.5\ {cm}^{3}\end{align*}$
::V=157.5厘米3

The answer is the volume of this triangular prism is $\begin{align*}157.5 \ cm^3\end{align*}$ .
::答案是三角棱镜的体积是157.5厘米3。

Example 5
::例5

What is the volume of a triangular prism with the following dimensions: $\begin{align*}b = 4 \ mm, \ h = 3 \ mm, \ H = 5 \ mm\end{align*}$
::b=4毫米, h=3毫米, H=5毫米

First, plug in the values for the area formula of a triangle to find the base area, $\begin{align*}B\end{align*}$ .
$\begin{aligned} B & = \frac{1}{2} b h \\ B & = \frac{1}{2} (4) (3) \\ B & = 2 (3) \\ B & = 6 m m^{2} \end{aligned}$
$\begin{align*}B & = \frac{1}{2} bh\\ B & = \frac{1}{2} (4) (3)\\ B & = 2 (3)\\ B & = 6 \ mm^2 \end{align*}$
::首先,插入三角形区域公式的值以找到基区域,B.B=12bhbhB=12(4)(3)(3)(3)B=2(3)B=6毫米2

Next, plug the triangle area value, $\begin{align*}B\end{align*}$ , and the value for the height of the prism, $\begin{align*}H\end{align*}$ , into the volume formula and multiply the values together.
$\begin{aligned} V & = B H \\ V & = 6 \times 5 \end{aligned}$
$\begin{align*}V & = BH\\ V & = 6 \times 5\\\end{align*}$
::下一步,在音量公式中插入三角区域值、B和棱晶高度的数值H,然后将数值乘在一起。V=BHV=6x5

$\begin{aligned} V & = 6 \times 5 \\ V & = 30 \end{aligned}$
$\begin{align*}V&= 6\ \times\ 5\\ V&= 30\end{align*}$
::V=6×5V=30 V=6×5V=30

Last, record the answer, making sure to include the appropriate unit of measurement.
::最后,记录答案,确保包括适当的计量单位。

$\begin{aligned} V = 30 {m m}^{3} \end{aligned}$
$\begin{align*}V = 30\ {mm}^{3}\end{align*}$
::V=30毫米3

The answer is the volume of this triangular prism is $\begin{align*}30 \ mm^3\end{align*}$ .
::答案是三角棱镜的体积是30毫米3

Review
::回顾

Find the volume of each triangular prism. Remember that $\begin{align*}h\end{align*}$ means the height of the triangular base and $\begin{align*}H\end{align*}$ means the height of the whole prism.
::查找每个三角棱柱的体积。记住, h 表示三角基点的高度, H 表示整个棱柱的高度。

$\begin{align*}b = 6 \ in, \ h = 4 \ in, \ H = 5 \ in\end{align*}$
::b=6 英寸, h=4 英寸, H=5 英寸

$\begin{align*}b = 7 \ in, \ h = 5 \ in, \ H = 9 \ in\end{align*}$
::b=7 英寸, h=5 英寸, H=9 英寸

$\begin{align*}b = 10 \ m, \ h = 8 \ m, \ H = 9 \ m\end{align*}$
::b=10米,h=8米,H=9米

$\begin{align*}b = 12 \ m, \ h = 10 \ m, \ H = 13 \ m\end{align*}$
::b=12米,h=10米,H=13米

$\begin{align*}b = 8 \ cm, \ h = 6 \ cm, \ H = 9 \ cm\end{align*}$
::b=8厘米,h=6厘米,H=9厘米

$\begin{align*}b = 9 \ cm, \ h = 7 \ cm, \ H = 8 \ cm\end{align*}$
::b=9厘米,h=7厘米,H=8厘米

$\begin{align*}b = 5.5 \ mm, \ h = 4 \ mm, \ H = 4 \ mm\end{align*}$
::b=5.5毫米, h=4毫米, H=4毫米

$\begin{align*}b = 11 \ cm, \ h = 9 \ cm, \ H = 8 \ cm\end{align*}$
::b=11厘米,h=9厘米,H=8厘米

$\begin{align*}b = 20 \ ft, \ h = 17 \ ft, \ H = 19 \ ft.\end{align*}$
::b=20英尺,h=17英尺,H=19英尺。

$\begin{align*}b = 20 \ ft, \ h = 18 \ ft, \ H = 15 \ ft.\end{align*}$
::b=20英尺,h=18英尺,H=15英尺。

$\begin{align*}b = 18 \ ft, \ h = 16 \ ft, \ H = 17 \ ft.\end{align*}$
::b=18英尺,h=16英尺,H=17英尺。

$\begin{align*}b = 24 \ ft, \ h = 21 \ ft, \ H = 19\ ft.\end{align*}$
::b=24英尺,h=21英尺,H=19英尺。

$\begin{align*}b = 24.5 \ ft, \ h = 18 \ ft, \ H = 16 \ ft.\end{align*}$
::b=24.5英尺,h=18英尺,H=16英尺。

$\begin{align*}b = 99 \ ft, \ h = 80 \ ft, \ H = 75 \ ft.\end{align*}$
::b=99英尺,h=80英尺,H=75英尺。

$\begin{align*}b = 100 \ ft, \ h = 80 \ ft, \ H = 110 \ ft.\end{align*}$
::b=100英尺,h=80英尺,H=110英尺。

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

Section outline

General

三角棱晶体体积

Finding the Volume of a Triangular Prism ::查找三角棱柱的量

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

Finding the Volume of a Triangular Prism
::查找三角棱柱的量

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