Course: 8.6 棱晶表面面积

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棱晶表面面积
Crystal has a fragile gift that she is packing in a box before wrapping it. To figure out how much wrapping paper she will need she has drawn the box with all its dimensions . How can Crystal calculate how much gift-wrap she will need using the above dimensions?

In this concept, you will learn to find the of prisms.
::在这个概念中,你会学会找到棱镜。

Surface Area
::地表地区

A prism is a three–dimensional figure with two parallel congruent polygons as bases. The side faces of a prism are rectangular in shape.
::棱晶是一个三维图,以两个平行的相似多边形作为基体。棱晶的侧面是长方形。

One measure when working with three–dimensional figures is called surface area . Surface area is the total of the areas of each face of a . Imagine you could wrap one of these figures in wrapping paper, like a present. The amount of wrapping paper needed to cover the figure represents its surface area. To find the surface area, you must be able to calculate the area of each face and then add these areas together.
::在使用三维图时,一个尺度被称作表面区域。表面区域是 a 中每个面的总面积。想象一下,您可以在包装纸中将其中的一个数字包起来,就像一个礼物一样。覆盖该图所需的包装纸数量代表着其表面区域。要找到表面区域,您必须能够计算每个面的面积,然后将这些区域加在一起。

There are several different ways to calculate surface area. One way is to use a . A net is a two-dimensional diagram of a three-dimensional figure. Imagine you could unfold a box so that it is completely flat. You would have something that looks like this.
::有几种不同的方法来计算表面积。一种方法是使用。网是一个三维图的二维图。想象一下你可以打开一个盒子, 使它完全平坦。你会有像这样的东西。

If you folded this up, you could see that it would form a cube. A cube is made up of faces that are squares. If you wanted to figure out the surface area or measurement of the outer covering of this cube, then you could find the area of each surface of the cube and then add the products together.
::如果您折叠了它, 您可以看到它会形成一个立方体。一个立方体是由方形的面孔组成。如果您想要找出这个立方体的表面面积或外部覆盖的测量, 那么您就可以找到立方体每个表面的面积, 然后将产品加在一起。

You could also look at a net of a rectangular prism .
::你也可以看一个长方形棱柱的网。

A rectangular prism is made up of . To find the surface area of a prism, you would need to calculate the area of each of the faces and then add them together.
::矩形棱镜由 . 组成。要找到棱镜的表面区域, 您需要计算每个面孔的面积, 然后将它们加在一起。

Let’s begin by calculating the surface area of a rectangular prism.
::让我们首先计算一个矩形棱柱的表面面积。

First, let’s break apart the shape so you can organize the six area calculations.
::首先,让我们拆分形状,以便您可以组织六个区域计算。

Next, calculate the area of each face.
::接下来,计算每个面孔的面积。

$\begin{aligned} \begin{array}{rcl} A_{bottom face} & = & l \times w \\ = & 12 \times 7 \\ = & 84 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A_{\text{bottom face}} &=& l \times w \\ &=& 12 \times 7 \\ &=& 84 \end{array}\end{align*}$
::下表层=lxw=12x7=84

$\begin{aligned} \begin{array}{rcl} A_{longer side} & = & l \times w \\ = & 12 \times 3 \\ = & 36 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A_{\text{longer side}} &=& l \times w \\ &=& 12 \times 3 \\ &=& 36 \end{array}\end{align*}$
::侧侧=lxw=12x3=36

$\begin{aligned} \begin{array}{rcl} A_{top face} & = & l \times w \\ = & 12 \times 7 \\ = & 84 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A_{\text{top face}} &=& l \times w \\ &=& 12 \times 7 \\ &=& 84 \end{array}\end{align*}$
::Atop face=lxw=12x7=84

$\begin{aligned} \begin{array}{rcl} A_{shorter side} & = & l \times w \\ = & 7 \times 3 \\ = & 21 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A_{\text{shorter side}} &=& l \times w \\ &=& 7 \times 3 \\ &=& 21 \end{array}\end{align*}$
::Ashorter 侧侧=lxw=7x3=21

$\begin{aligned} \begin{array}{rcl} A_{longer side} & = & l \times w \\ = & 12 \times 3 \\ = & 36 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A_{\text{longer side}} &=& l \times w \\ &=& 12 \times 3 \\ &=& 36 \end{array}\end{align*}$
::侧侧=lxw=12x3=36

$\begin{aligned} \begin{array}{rcl} A_{shorter side} & = & l \times w \\ = & 7 \times 3 \\ = & 21 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A_{\text{shorter side}} &=& l \times w \\ &=& 7 \times 3 \\ &=& 21 \end{array}\end{align*}$ Then, add the areas together.
::Ashorter 侧侧=lxw= 7x3= 21, 然后将区域加在一起。

$\begin{aligned} \begin{array}{rcl} Surface Area & = & 84 + 84 + 36 + 36 + 21 + 21 \\ Surface Area & = & 282 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{Surface Area} &=& 84+84+36+36+21+21 \\ \text{Surface Area} &=& 282 \end{array}\end{align*}$
::表面面积=84+84+84+36+36+36+21+21+21表面面积=282

The answer is 282.
::答案是282

The surface area of the prism is $\begin{align*}282 \ in^2\end{align*}$ .
::棱晶表面面积为282英寸2。

Now you can see each face so that you can calculate their area and then add them together. However, you can also use a formula to represent the faces as you find their area. The formula gives a nice short cut that you can use for any kind of prism, no matter what shape its base is.
::现在,您可以看到每张脸,这样您就可以计算出它们的区域,然后将它们相加。然而,您也可以在找到它们的区域时使用一个公式来代表它们的脸。公式提供了一个很好的捷径,可以用于任何类型的棱镜,不管其底部是什么形状。

Take a look at the formula below.
::看看下面的公式。

$\begin{aligned} S A = P h + 2 B \end{aligned}$
$\begin{align*}SA=Ph+2B\end{align*}$
::SA=Ph+2B

Let’s look at the first part of the formula. $\begin{align*}P\end{align*}$ represents the perimeter of the base, and $\begin{align*}h\end{align*}$ represents the height of the prism. By multiplying the perimeter and height, you are finding the area of all of the side faces at once. This will be very useful if the prism that you are working with isn’t just a cube or a rectangular prism.
::让我们看看公式的第一部分。 P 代表基底的周边, h 代表棱柱的高度。通过将周边和高度乘以,您可以同时找到侧面所有面孔的区域。如果您使用的棱柱不仅仅是立方体或矩形棱柱,这将非常有用。

The second part of the formula represents the area of the top and bottom faces. $\begin{align*}B\end{align*}$ represents the area of one base, which you find using whichever area formula is appropriate for the shape of the base. Then you multiply it by 2 to show the area of the top and bottom faces at once.
::公式的第二部分代表上面和下面的面积。 B 代表一个基的面积, 使用哪个区域公式适合基的形状。然后乘以 2 以同时显示上面和下面的面积。

Let’s look at an example.
::让我们举个例子。

Find the surface area of this figure using a formula.
::使用公式查找此图的表面区域。

First, find the perimeter of the base.
::首先,找到基地的周边

$\begin{aligned} \begin{array}{rcl} P & = & 2 l + 2 w \\ P & = & 2 \times 21 + 2 \times 14 \\ P & = & 70 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P &=& 2l+2w \\ P &=& 2 \times 21+2 \times 14\\ P &=& 70 \end{array}\end{align*}$
::P=2l+2wP=2x21+2x14P=70

Next, calculate the area of the base.
::下一步,计算基数的面积。

$\begin{aligned} \begin{array}{rcl} A & = & l \times w \\ A & = & 21 \times 14 \\ A & = & 294 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& l \times w \\ A &=& 21 \times 14 \\ A &=& 294 \end{array}\end{align*}$
::A=lxwA=21x14A=294

Then, knowing the height is 5 cm, determine the surface area.
::然后,知道高度是5厘米, 确定表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & P h + 2 B \\ S A & = & 70 \times 5 + 2 \times 294 \\ S A & = & 350 + 588 \\ S A & = & 938 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& Ph+2B \\ SA &=& 70 \times 5+2 \times 294 \\ SA &=& 350+588 \\ SA &=& 938 \end{array}\end{align*}$
::SA=Ph+2BSA=70x5+2x294SA=350+5888SA=938

The answer is 938.
::答案是938

The surface area of the prism is $\begin{align*}938 \ cm^2\end{align*}$ .
::棱晶表面面积为938厘米2。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Crystal and her gift box.
::早些时候,你得到一个问题关于Crystal和她的礼物盒。

The box to be wrapped has a length of $\begin{align*}12^{\prime \prime}\end{align*}$ , a wide of $\begin{align*}9^{\prime \prime}\end{align*}$ and a height of $\begin{align*}6^{\prime \prime}\end{align*}$ .
::要包装的盒子长度为 12 ,宽度为 9 ,高度为 6 。

First, find the perimeter of the base.
::首先,找到基地的周边

$\begin{aligned} \begin{array}{rcl} P & = & 2 l + 2 w \\ P & = & 2 \times 12 + 2 \times 9 \\ P & = & 42 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P &=& 2l+2w \\ P &=& 2 \times 12+2 \times 9 \\ P &=& 42 \end{array}\end{align*}$
::P=2l+2wP=2x12+2x9P=42

Next, calculate the area of the base.
::下一步,计算基数的面积。

$\begin{aligned} \begin{array}{rcl} A & = & l \times w \\ A & = & 12 \times 9 \\ A & = & 108 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& l \times w \\ A &=& 12 \times 9\\ A &=& 108 \end{array}\end{align*}$
::A=lxwA=12x9A=108

Then, knowing the height is 6 in., determine the surface area.
::然后,知道高度是6英寸, 确定表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & P h + 2 B \\ S A & = & 42 \times 6 + 2 \times 108 \\ S A & = & 252 + 216 \\ S A & = & 468 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& Ph+2B \\ SA &=& 42 \times 6+2 \times 108 \\ SA &=& 252+216 \\ SA &=& 468 \end{array}\end{align*}$
::SA=Ph+2BSA=42x6+2x108SA=252+216SA=468

The answer is 468.
::答案是468。

Crystal needs $\begin{align*}468 \ in.^2\end{align*}$ of wrapping paper.
::Crystal需要包装纸的468英寸2。

Example 2
::例2

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

First, find the perimeter of the base.
::首先,找到基地的周边

$\begin{aligned} \begin{array}{rcl} P & = & 2 a + b \\ P & = & 2 \times 5 + 8 \\ P & = & 18 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P &=& 2a+b \\ P &=& 2 \times 5+8 \\ P &=& 18 \end{array}\end{align*}$
::P=2a+bP=2x5+8P=18

Next, calculate the area of the base.
::下一步,计算基数的面积。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} 8 \times 3 \\ A & = & 12 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2}bh \\ A &=& \frac{1}{2} 8 \times 3 \\ A &=& 12 \end{array}\end{align*}$
::A=12bhA=128x3A=12

Then, knowing the height is 15 in, determine the surface area.
::然后,知道高度是15英寸, 确定表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & P h + 2 B \\ S A & = & 18 \times 15 + 2 \times 12 \\ S A & = & 270 + 24 \\ S A & = & 294 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& Ph+2B \\ SA &=& 18 \times 15+2 \times 12 \\ SA &=& 270+24 \\ SA &=& 294 \end{array}\end{align*}$
::SA=Ph+2BSA=18x15+2x12SA=270+24SA=294

The answer is 294.
::答案是294

The surface area of the prism is $\begin{align*}294 \ in^2\end{align*}$ .
::棱晶表面面积为294英寸2。

Example 3
::例3

True or false: The surface area includes the inside of a prism.
::真实的或虚假的:表面区域包括棱镜的内侧。

The surface area is the measurement of the outer covering of a prism.
::表面区域是对棱晶外层的测量。

The answer is false.
::答案是虚假的。

Example 4
::例4

True or false: A net shows all three dimensions of a prism.
::真实的或假的: 网显示棱镜的所有三个维度。

A net does show all dimensions of a prism.
::网确实显示了棱镜的所有维度。

The answer is true.
::答案是真实的。

Example 5
::例5

True or false: You know a figure is a prism because the faces are rectangles.
::真实的或假的:你知道一个数字是棱镜,因为脸部是矩形。

A figure with all rectangular faces is a prism.
::带有所有长方形面的图是一个棱镜。

The answer is true.
::答案是真实的。

Review
::回顾

Look at each figure and then answer the following questions.
::看看每一图,然后回答下列问题。

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

2. What is the surface area of this figure?
::2. 这一数字的表面积是多少?

3. What is the shape of it’s base?
::3. 基地是什么形状?

4. What is the area of this figure’s large base?
::4. 这个数字的庞大基地有哪些领域?

5. What is the name of this figure?
::5. 这个数字叫什么名字?

6. What is the surface area of this figure?
::6. 这一数字的表面积是多少?

7. What is the shape of it’s base?
::7. 基地的形状是什么?

8. What is the height of this figure?
::8. 这个数字的高度是多少?

9. What is the area of this figure’s base?
::9. 这个数字的基数有哪些领域?

10. What is the name of this figure?
::10. 这个数字叫什么名字?

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

12. How many bases does this figure have?
::12. 这一数字有多少基础?

13. How many side faces are there?
::13. 有多少面孔?

14. What is the surface area of this figure?
::14. 这一数字的表面积是多少?

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

棱晶表面面积

Surface Area ::地表地区

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

Surface Area
::地表地区

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