Section: 棱镜表面积 | 10.14棱镜表面积

Section outline

Leola owns an upholstery shop that specializes in refurbishing old furniture. A customer brought in an old wooden chest and wants Leola to cover the entire chest in a new vinyl covering. Leola needs to find the total of the chest so she can order the vinyl covering. The chest is a rectangular prism with a pictured below.
::利奥拉拥有一家专门翻修旧家具的室内装饰店。一位顾客带进一个旧木箱,希望利奥拉用新的乙烯盖覆盖整个胸部。莱奥拉需要找到整个胸部,以便她可以订购乙烯盖。胸部是一个长方形棱镜,下面有图画。

How can Leola figure out the total surface area of the chest?
::莉奥拉怎么才能发现胸口的表面面积呢?

In this concept, you will learn to identify the surface area of prisms by using nets and formulas.
::在这个概念中,你将学会通过使用网和公式来确定棱晶的表面区域。

Finding Surface Area of Prisms
::棱晶表面地区

You can compute the area of a two-dimensional flat plane , like the rectangle below, by multiplying the length times the width.
::您可以通过乘以宽度的倍数来计算二维平面的面积,如下面的矩形。

This rectangle has a length of 8 inches and a width of 3 inches.
::这个矩形长8英寸,宽3英寸。

$\begin{aligned} 8 i n . \times 3 i n . = 24 s q . i n \end{aligned}$
::8 i n. × 3 i n. = 24 s q. i n

The area of the rectangle is $\begin{aligned} 24 s q . i n \end{aligned}$ or $\begin{aligned} i n^{2} \end{aligned}$ .
::矩形区域为 24 sq . i n 或 i n 2。

When it comes to , you do not just calculate the area, you have to calculate the total surface area , which is the total area of each of the faces of a three-dimensional object. Let’s look at a cube and see how this works.
::当谈到时, 您不仅计算区域, 您必须计算总表面区域, 即三维天体每个表面的总面积。让我们看看一个立方体, 看看它是如何工作的。

The surface area of this cube would be the total area of all of its surfaces. Finding the surface area of a figure is very useful when painting or covering a three-dimensional solid.
::此立方体的表面积将是其所有表面的总面积。找到一个图的表面积在绘制或覆盖三维固体时非常有用。

To figure out the surface area, calculate the area of each of the faces of the solid and then add all of the areas for the total surface area. Sometimes, it is hard to see all the sides of a solid object, so a net is used to see all the sides of a three-dimensional solid.
::解析表面区域, 计算固体每个面的面积, 然后将全部面积添加到全部表面区域。有时, 很难看到一个固体物体的两面, 所以用一个网来查看三维固体的两面。

A net is a drawing that shows a “flattened out” picture of the solid. With a net, you can see each part of the solid. If you were to make a net out of paper and fold it up, you would be able to create a .
::网是一张显示固体“减缩”图片的图纸。用一个网,你可以看到固体的每个部分。如果你用纸做成一个网,把它折叠起来,你就可以创建一个。

Here is a net of a cube.
::这是一块立方体的网。

A cube has six faces – two bases and four sides. Well, you can see here that this net also has six faces. If you were to fold this figure using the line segments, it would create a cube.
::立方体有六张面孔 — — 两个基点和四边。您可以在这里看到这个网也拥有六张面孔。如果您使用线段折叠这个数字, 它将会创建立方体。

To calculate surface area of a cube, find the area of each of the six faces and then add them all to get the surface area.
::要计算一个立方体的表面面积, 找到六张脸的每个区域, 然后添加它们来获取表面区域。

You can use the formula for finding the area of a square to find the area of one square. The length of one of the sides of the square face is 3 inches.
::您可以使用公式找到一个正方的区域以找到一个正方的区域。正方面的一边长度为 3 英寸。

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & 3^{2} \\ A & = & 9 s q . i n . \end{array} \end{aligned}$

This is the area for one square face. There are six square faces. If you take this area and multiply it by six, you will have the surface area of the cube.
::这是一张正方形的面积。有六张正方形的面积。如果您将这个区域乘以 6, 您将拥有立方体的表面积。

$\begin{aligned} \begin{array}{rcl} 9 (6) & = & 54 s q . i n . \\ 9 + 9 + 9 + 9 + 9 + 9 & = & 54 i n .^{2} \end{array} \end{aligned}$

The surface area of the cube is $\begin{aligned} 54 i n^{2} \end{aligned}$ .
::立方体的表面面积为54 i n 2 。

Now let’s look at a rectangular prism net.
::现在让我们来看看长方形棱镜网。

Start by finding the area of each rectangle.
::首先找到每个矩形的区域

There are two long faces, two short faces, one bottom, and one top that matches it.
::有两个长长的面孔,两个短的面孔,一个底部,一个顶部相匹配。

First, let’s find the area of the bottom. It has a length of $\begin{aligned} 6^{''} \end{aligned}$ and a width of $\begin{aligned} 4^{''} \end{aligned}$ . Since the shape of the bottom is a rectangle, you can use the formula for finding the area of a rectangle.
::首先,让我们找到底部的面积。底部的长度为 6 和 4 。由于底部的形状是一个矩形, 您可以使用公式来找到矩形的面积。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (6) (4) \\ A & = & 24 s q . i n . \end{array} \end{aligned}$

Since the top and bottom match, multiply this area by two: $\begin{aligned} 24 \times 2 = 48 square inches \end{aligned}$ .
::自上方和下方匹配后,此区域乘以 2: 24× 2 = 48 平方英寸。

Next, find the area of the two long faces. Each long face is a rectangle in shape. The length is $\begin{aligned} 6^{''} \end{aligned}$ and the width is $\begin{aligned} 2^{''} \end{aligned}$ .
::下一步, 找到两个长面的区域。每个长面都是一个矩形。长度为 6 , 宽度为 2 。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (6) (2) \\ A & = & 12 s q . i n . \end{array} \end{aligned}$

Since there are two long faces to the prism , take this measure and multiply it by two.
::由于棱镜有两个长长的面孔,采取这一尺度,再乘以二。

$\begin{aligned} A = 24 s q . i n \end{aligned}$

Next, find the area of the two shorter faces. Each face is a small rectangle. The length is $\begin{aligned} 4^{''} \end{aligned}$ and the width is $\begin{aligned} 2^{''} \end{aligned}$ .
::接下来,找到两个较短面孔的面积。每个面孔是一个小矩形。长度是 4 ,宽度是 2 。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (4) (2) \\ A & = & 8 s q . i n . \end{array} \end{aligned}$

Since there are two short faces to the prism, take this measure and multiply it by two.
::因为棱镜有两个短长的面孔,所以采取这一尺度,再乘以二。

$\begin{aligned} \begin{array}{rcl} A & = & 2 (8) \\ A & = & 16 s q . i n . \end{array} \end{aligned}$

To find the surface area of the entire prism, add up the areas of all of the faces.
::要找到整个棱晶的表面区域, 将所有面孔的面积加在一起。

$\begin{aligned} S A = 16 + 24 + 48 = 88 s q . i n . \end{aligned}$

The answer is 88 sq. inches.
::答案是88平方英寸

A triangular prism is a prism with two parallel congruent bases that are triangles. The faces of the prism are , but the bases are triangles. Here is a picture of a triangular prism.
::三角棱镜是一个棱镜,它有两个平行的相似基点,它们是三角形。棱镜的面部是,但底部是三角形。这里是三角棱镜的图片。

Here is what the net of a triangular prism would look like.
::这就是三角棱镜的网状。

To find the surface area, you need to figure out the area of the bottom, right side, left side and two bases which are triangles.
::要找到表面积, 您需要找出底部、右侧、左侧和两个三角基座的面积。

The bottom is a rectangle. It has a length of 7 cm and a width of 3 cm.
::底部是矩形,长度为7厘米,宽度为3厘米。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 7 (3) \\ A & = & 21 s q . c m . \end{array} \end{aligned}$

The left side is a rectangle. It has a length of 7 cm and a width of 4 cm.
::左侧是一个矩形,长7厘米,宽4厘米。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 7 (4) \\ A & = & 28 s q . c m . \end{array} \end{aligned}$

The right side is a rectangle. It has a length of 7 cm and a width of 5 cm.
::右侧是一个矩形,长7厘米,宽5厘米。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 7 (5) \\ A & = & 35 s q . c m . \end{array} \end{aligned}$

The bases are two triangles. They have a base of 3 cm and a height of 4 cm.
::基座是两个三角形,基座3厘米,高4厘米。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (3) (4) \\ A & = & \frac{1}{2} (12) \\ A & = & 6 s q . c m \end{array} \end{aligned}$

There are two triangles, so multiply this base by two.
::有两个三角形, 所以将这个基数乘以 2 。

$\begin{aligned} \begin{array}{rcl} A & = & 2 (6) \\ A & = & 12 s q . c m . \end{array} \end{aligned}$

Now, add all of the areas.
::现在,添加所有区域。

$\begin{aligned} \begin{array}{rcl} S A & = & 12 + 35 + 28 + 21 \\ S A & = & 96 s q . c m . \end{array} \end{aligned}$

The answer is 96 sq. cm.
::答案是96平方厘米

Let’s look at another example.
::让我们再看看另一个例子。

To find the surface area of this rectangular prism, you have to figure out the sum of all of the areas. Here is a formula to do this.
::要找到这个矩形棱柱的表面区域, 您必须找出所有区域的和和。这是要做到这一点的公式。

$\begin{aligned} S A = 2 (l w + l h + w h) \end{aligned}$

Substitute the given values into the formula. The length of the prism is 9 inches, the width is 3 inches and the height is 5 inches.
::将给定值替换为公式。棱晶长度为 9 英寸, 宽度为 3 英寸, 高度为 5 英寸。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 (9 (3) + 9 (5) + (3) 5) \\ S A & = & 2 (27 + 45 + 15) \\ S A & = & 2 (87) \\ S A & = & 174 s q . i n . \end{array} \end{aligned}$

You can use this same method with a triangular prism. Let’s look at an example.
::您可以用三角棱镜使用同样的方法。让我们来举个例子。

$\begin{aligned} S A = Area of three rectangles + Area of two triangles \end{aligned}$

Substitute the given values into the formula and solve.
::将给定值替换为公式和解析。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 (8 + 9 + 7) + 2 (\frac{1}{2} (8) 7) \\ S A & = & 2 (24) + 2 (28) \\ S A & = & 48 + 56 \\ S A & = & 104 s q . i n . \end{array} \end{aligned}$

The answer is 104 sq. in.
::答案是104平方英寸英寸

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Leola and the chest she is upholstering.
::早些时候,你得到一个问题关于Leola 和她胸口她正在装饰。

Leola needs to know the total surface area of the chest so she can order the covering. The net is pictured below.
::Leola需要知道胸口的表面总面积,以便她可以订购覆盖物。网图如下。

The chest’s measurements are $\begin{aligned} 24^{''} \end{aligned}$ long $\begin{aligned} \times 8^{''} \end{aligned}$ wide by $\begin{aligned} 18^{''} \end{aligned}$ high.
::胸腔的测量量为24 × 长 × 8 × 宽 18 高。

To figure out the surface area of the chest, Leola can use the formula for finding the surface area of a rectangular prism.
::为了找出胸口的表面面积,Leola可以使用公式找到矩形棱柱的表面面积。

$\begin{aligned} S A = 2 (l w + l h + w h) \end{aligned}$

First, Leola needs to substitute the given information into the formula.
::首先,Leola需要将给定的信息换成公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 (24 (18) + 24 (18) + (18) 18) \\ S A & = & 2 (432 + 432 + 324) \\ S A & = & 1188 square inches \end{array} \end{aligned}$

Leola will need to purchase 1188 square inches of material.
::Leola需要购买1 188平方英寸的材料。

Example 2
::例2

Find the surface area of the cube.
::找到立方体的表面区域。

To solve this problem, you can use the formula for surface area.
::要解决这个问题,您可以使用表面积的公式。

$\begin{aligned} A = s^{2} \end{aligned}$

First, substitute the given value into the formula. The length of one of the sides of the square face is 4 inches.
::首先, 将给定值替换为公式。方形两侧的长度为 4 英寸。

$\begin{aligned} \begin{array}{rcl} A & = & 4^{2} \\ A & = & 16 s q . i n . \end{array} \end{aligned}$

This is the area for one square face. There are six square faces.
::这是一张正方形脸的区域。有六张正方形脸。

Next, multiply 16 sq. in. by six.
::下一个乘以16乘以6 乘以16乘以6

$\begin{aligned} 16 (6) = 96 s q . i n . \end{aligned}$

The surface area of the cube is $\begin{aligned} 96 i n^{2} \end{aligned}$ .
::立方体的表面面积为96 i n 2 。

Example 3
::例3

Find the surface area of the prism.
::寻找棱镜的表面区域。

To find the surface area of this rectangular prism, use the following formula.
::要找到此矩形棱柱的表面区域, 请使用以下公式。

$\begin{aligned} S A = 2 (l w + l h + w h) \end{aligned}$

First, substitute the given values into the formula. The length of the prism is 15 inches, the width is 5 inches and the height is 8 inches.
::首先, 将给定值替换为公式。棱晶长度为 15 英寸, 宽为 5 英寸, 高度为 8 英寸。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 (15 (5) + 15 (8) + (5) 8) \\ S A & = & 2 (75 + 120 + 40) \\ S A & = & 2 (235) \\ S A & = & 470 s q . i n . \end{array} \end{aligned}$

The answer is 470 sq. inches.
::答案是470平方英寸

Example 4
::例4

Find the surface area of the prism.
::寻找棱镜的表面区域。

First, figure out the areas of the bottom, right side, left side and two bases which are triangles.
::首先,找出底部、右侧、左侧和两个三角基座的区域。

The bottom is a rectangle. It has a length of 6 m and a width of 4 m.
::底部是一个矩形,长6米,宽4米。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 6 (4) \\ A & = & 24 s q . m . \end{array} \end{aligned}$

The left side is a rectangle. It has a length of 6 m and a width of 3 m.
::左侧是一个矩形,长6米,宽3米。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 3 (6) \\ A & = & 18 s q . m . \end{array} \end{aligned}$

The right side is a rectangle. It has a length of 6 m and a width of 3 m.
::右侧是一个矩形,长6米,宽3米。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 6 (3) \\ A & = & 18 s q . m . \end{array} \end{aligned}$

The bases are two triangles. They have a base of 4 m and a height of 3 m.
::基座是两个三角形,基座4米,高度3米。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (4) (3) \\ A & = & \frac{1}{2} (12) \\ A & = & 6 s q . m \end{array} \end{aligned}$

There are two triangles, so multiply this base by two.
::有两个三角形, 所以将这个基数乘以 2 。

$\begin{aligned} \begin{array}{rcl} A & = & 2 (6) \\ A & = & 12 s q . m . \end{array} \end{aligned}$

Next, add all of the areas.
::下面加上所有区域。

$\begin{aligned} \begin{array}{rcl} S A & = & 24 + 18 + 18 + 12 \\ S A & = & 72 s q . m . \end{array} \end{aligned}$

The answer is 72 sq. meters.
::答案是72平方米

Example 5
::例5

Find the surface area of the prism.
::寻找棱镜的表面区域。

To find the surface area of this rectangular prism, you have to figure out the sum of all of the areas. Here is a formula to do this.
::要找到这个矩形棱柱的表面区域, 您必须找出所有区域的和和。这是要做到这一点的公式。

$\begin{aligned} S A = 2 (l w + l h + w h) \end{aligned}$

First, substitute the given values into the formula. The length of the prism is 1 cm, the width is 2 cm, and the height is 13 cm.
::首先, 将给定值替换为公式。棱晶长度为 1 厘米, 宽度为 2 厘米, 高度为 13 厘米。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 (1 (2) + 1 (13) + (2) 13) \\ S A & = & 2 (2 + 13 + 26) \\ S A & = & 2 (41) \\ S A & = & 82 s q . c m \end{array} \end{aligned}$

The answer is 82 sq. cm.

::答案是82平方米

Review
::回顾

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。

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。

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。

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。

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。

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。

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。

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。

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。

Use the following figure to answer each question.
::使用下图回答每个问题。

What unit are the measurements of this figure?
::这个数字的测量单位是什么?

What is the length of the base?
::基数的长度是多少?

What is the width of the base?
::基地的宽度是多少?

What is the measure of the triangular side length?
::三角边长度的度量是多少?

What is the height of the triangular face?
::三角形脸的高度是多少?

What is the formula for finding the surface area of a triangular prism?
::找到三角棱镜表面积的公式是什么?

What is the surface area of this figure?
::这个数字的表面面积是多少?

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

Finding Surface Area of Prisms ::棱晶表面地区

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

Finding Surface Area of Prisms
::棱晶表面地区

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