节: 圆柱体表面面积 | 8.7 气瓶表面面积

章节大纲

Mrs. Johnson is wrapping a cylindrical package in brown paper so that she can mail it to her son. Based on the dimensions shown, how much paper will she need to cover the package?
::Johnson夫人用棕色纸包裹一个圆柱形包裹,以便她可以把它寄给儿子。

In this concept, you will learn to find the of cylinders.
::在这个概念中,你会学会找到圆柱体。

Surface Area
::地表地区

A cylinder has two parallel congruent circular bases with a curved rectangle as its side. One way is to use a .
::一个圆柱体有两个平行的同步圆环基,其侧面是一个曲线矩形。一种方法是使用。

A net is a two-dimensional diagram of a three-dimensional figure. If you could unfold a cylinder (like a can) so that it is completely flat, you would have something that looks like this.
::网是一个三维图的二维图。如果您可以打开一个圆柱形( 如一个罐子) , 使其完全平坦, 您就会有类似这样的东西。

With the net of a cylinder, you would need to calculate the area of each circle and the area of the curved side of the cylinder. Then you could add these values together to find the surface area.
::使用圆柱形网,您需要计算圆形每个圆形的面积和圆柱形曲线侧的面积。然后您可以将这些数值加在一起,以找到表面区域。

The formula $\begin{aligned} A = π r^{2} \end{aligned}$ can be used to find the area of a circle bases. Look closely again at the cylinder above. The two circular faces are congruent, so they must have the same radius and diameter .
::Ar2 公式可用于查找圆基的面积。仔细查看上面的圆柱体。两个圆圆面是相同的, 其半径和直径必须相同。

Let’s look at an example.
::让我们举个例子。

First, calculate the area of the circular bases.
::首先,计算圆形基地的区域。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & π (4)^{2} \\ A & = & π (16) \\ A & = & 50.3 \end{array} \end{aligned}$

::2A(4)2A(16)A=50.3

Next, find the length of the cylinder’s side. You know the width is 8 cm but the length is not shown. The cylinder was unwrapped to form the net. Therefore, the of the circle would be the length of the side.
::下一步, 找到圆柱体侧的长度。您知道宽度是 8 厘米, 但是没有显示长度。圆柱体被解开以形成网状。因此, 圆圈的长度将是侧面的长度。

$\begin{aligned} \begin{array}{rcl} C & = & 2 π r \\ C & = & 2 π \times 4 \\ C & = & 25.1 \end{array} \end{aligned}$

::C=2rC=24C=25.1

Then, calculate the area of the side.
::然后计算侧面的区域。

$\begin{aligned} \begin{array}{rcl} A & = & l \times w \\ A & = & 25.1 \times 8 \\ A & = & 200.8 \end{array} \end{aligned}$

::A=lxwA=25.1x8A=200.8

Then, find the surface area of the cylinder by adding the side area to the top and bottom area.
::然后,通过将侧区域加到顶部和底部区域,找到圆筒的表面区域。

$\begin{aligned} \begin{array}{rcl} S A & = & bottom + top + side \\ S A & = & 50.3 + 50.3 + 200.8 \\ S A & = & 301.4 \end{array} \end{aligned}$

::SA=底部+顶部+顶部SA=50.3+50.3+200.8SA=301.4

The answer is 301.4.
::答案是301.4。

The surface area of the cylinder is $\begin{aligned} 301.4 c m^{2} \end{aligned}$ .
::气瓶表面面积为301.4厘米2。

Putting this all together, you can use the following formula to find the surface area of the cylinder:
::将这些全部组合在一起,您可以使用以下公式来找到气瓶的表面区域:

$\begin{aligned} S A = 2 π r^{2} + 2 π r h \end{aligned}$

::SA=2r2+2rh

The formula $\begin{aligned} 2 π r^{2} \end{aligned}$ represents the area of the top and bottom circles of the cylinder. The $\begin{aligned} 2 π r h \end{aligned}$ represents the perimeter $\begin{aligned} (2 π r) \end{aligned}$ multiplied by the height , $\begin{aligned} h \end{aligned}$ .
::公式 2°r2 表示圆柱体顶部和底部圆圈的区域。 2°rh 表示周界(2°r)乘以高度, h.

Let’s look at an example.
::让我们举个例子。

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

First, substitute what you know into the surface area formula.
::首先,将您所知道的替换为表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π r^{2} + 2 π r h \\ S A & = & 2 π (3.5)^{2} + 2 π (3.5) (28) \end{array} \end{aligned}$

::SA=2r2+2rhSA=2(3.5)2+2(3.5)(28)

Next, calculate the surface area.
::接下来,计算表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π (3.5)^{2} + 2 π (3.5) (28) \\ S A & = & 2 π (12.25) + 2 π (98) \\ S A & = & 76.97 + 615.75 \\ S A & = & 692.72 \end{array} \end{aligned}$

::SA=2(3.5)2+2(3.5)(28)SA=2(12.25)+2(98)SA=76.97+615.75SA=692.72

The answer is 692.72.
::答案是692.72。

The surface area of the cylinder is $\begin{aligned} 692.7 c m^{2} \end{aligned}$ .
::气瓶的表面面积为692.7厘米。

Sometimes, you can have a cylinder that has been cut. This is called a truncated cylinder . This is where you only see a section of the cylinder and will need to figure out the surface area of what you see.
::有时候,你可以有一个被切割的圆柱体。这叫做短球体。这里你只看到圆柱体的某一部分,需要找出你所看到的东西的表面区域。

Now, let’s say that one half of the cylinder is pictured.
::现在,让我们假设,有一半的气瓶被想象出来。

First, substitute what you know into the surface area formula.
::首先,将您所知道的替换为表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π r^{2} + 2 π r h \\ S A & = & 2 π (2)^{2} + 2 π (2) (4) \end{array} \end{aligned}$

::SA=2r2+2rhSA=2(2)+2(2)(4)

Next, calculate the surface area.
::接下来,计算表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π (2)^{2} + 2 π (2) (4) \\ S A & = & 2 π (4) + 2 π (8) \\ S A & = & 25.1 + 50.3 \\ S A & = & 75.4 \end{array} \end{aligned}$

::SA=2(2)2+2(2)(4)SA=2(4)+2(8)SA=25.1+50.3SA=75.4

Then, the truncated cylinder is half a cylinder. Find half of the surface area.
::然后,截断的圆柱体是半个圆柱, 找到半个表面区域。

$\begin{aligned} \begin{array}{rcl} S A_{full cylinder} & = & 75.4 \\ S A_{half cylinder} & = & \frac{75.4}{2} \\ S A_{half cylinder} & = & 37.7 \end{array} \end{aligned}$

::半汽瓶=75.4SA 半汽瓶=75.42SA 半汽瓶=37.7

The answer is 37.7.
::答案是37.7

The surface area of the half-cylinder (truncated cylinder) is $\begin{aligned} 37.7 c m^{2} \end{aligned}$ .
::半圆柱体(圆柱体)的表面面积为37.7厘米2。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Mrs. Johnson’s cylindrical wrapping.
::更早之前, Johnson夫人的圆柱包装给您带来了一个问题。

First, you need to find the radius. Remember, the radius is half the measure of the diameter.
::首先,你需要找到半径。记住,半径是直径的一半。

$\begin{aligned} \begin{array}{rcl} r & = & \frac{d}{2} \\ r & = & \frac{11}{2} \\ r & = & 5.5 \end{array} \end{aligned}$

::= rd2r= 112r= 5.5

Next, substitute what you know into the surface area formula.
::接下来,将您所知道的替换为表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π r^{2} + 2 π r h \\ S A & = & 2 π (5.5)^{2} + 2 π (5.5) (22) \end{array} \end{aligned}$

::SA=2r2+2rhSA=2(5.5)2+2(5.5)(22)

Then, calculate the surface area.
::然后计算表面积

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π (5.5)^{2} + 2 π (5.5) (22) \\ S A & = & 2 π (30.25) + 2 π (121) \\ S A & = & 190.1 + 760.3 \\ S A & = & 950.4 \end{array} \end{aligned}$

::SA=2(5.5)2+2(5.5)(22)SA=2(30.25)+2(121)SA=190.1+760.3SA=950.4

The answer is 950.4.
::答案是950.4。

Mrs. Johnson needs $\begin{aligned} 950.4 c m^{2} \end{aligned}$ of brown paper to wrap her parcel.
::Johnson夫人需要950.4厘米的棕色纸来包裹包裹

Example 2
::例2

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

First, you need to find the radius. Remember that the radius is half the measure of the diameter.
::首先,您需要找到半径。记住半径是直径的半径的一半。

$\begin{aligned} \begin{array}{rcl} r & = & \frac{d}{2} \\ r & = & \frac{13}{2} \\ r & = & 6.5 \end{array} \end{aligned}$

Next, substitute what you know into the surface area formula.
::接下来,将您所知道的替换为表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π r^{2} + 2 π r h \\ S A & = & 2 π (6.5)^{2} + 2 π (6.5) (11) \end{array} \end{aligned}$

::SA=2r2+2rhSA=2(6.5)2+2(6.5)(11)

Then, calculate the surface area.
::然后计算表面积

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π (6.5)^{2} + 2 π (6.5) (11) \\ S A & = & 2 π (42.25) + 2 π (71.5) \\ S A & = & 265.5 + 449.2 \\ S A & = & 714.7 \end{array} \end{aligned}$

::SA=2(6.5)2+2(6.5)(11)SA=2(42.25)+2(71.5)SA=265.5+449.2SA=714.7

The answer is 714.7.
::答案是714.7。

The surface area of the cylinder is $\begin{aligned} 714.7 f t^{2} \end{aligned}$ .
::气瓶表面面积为714.7平方英尺。

Example 3
::例3

Find the surface area of a cylinder with a radius of 6 inches and a height of 5 inches.
::找到圆柱体的表面面积,半径为6英寸,高度为5英寸。

First, substitute what you know into the surface area formula.
::首先,将您所知道的替换为表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π r^{2} + 2 π r h \\ S A & = & 2 π (6)^{2} + 2 π (6) (5) \end{array} \end{aligned}$

::SA=2r2+2rhSA=2(6)-2+2(6)(5)

Next, calculate the surface area.
::接下来,计算表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π (6)^{2} + 2 π (6) (5) \\ S A & = & 2 π (36) + 2 π (30) \\ S A & = & 226.2 + 188.5 \\ S A & = & 414.7 \end{array} \end{aligned}$

::SA=226.2+188.5SA=414.7

The answer is 414.7.
::答案是414.7。

The surface area of the cylinder is $\begin{aligned} 414.7 i n^{2} \end{aligned}$ .
::气瓶表面面积为414.7英寸2。

Example 4
::例4

Find the surface area of a cylinder with a radius of 4 cm and a height of 12 cm.
::找到圆柱体的表面面积,半径为4厘米,高度为12厘米。

First, substitute what you know into the surface area formula.
::首先,将您所知道的替换为表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π r^{2} + 2 π r h \\ S A & = & 2 π (4)^{2} + 2 π (4) (12) \end{array} \end{aligned}$

::SA=2r2+2rhSA=2(4)2+2(4)(4)(12)

Next, calculate the surface area.
::接下来,计算表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π (4)^{2} + 2 π (4) (12) \\ S A & = & 2 π (16) + 2 π (48) \\ S A & = & 100.5 + 301.6 \\ S A & = & 402.1 \end{array} \end{aligned}$

::SA=2(4)2+2(4)(12)(12)SA=2(16)+2(48)SA=100.5+301.6SA=402.1

The answer is 402.1.
::答案是402.1

The surface area of the cylinder is $\begin{aligned} 402.1 c m^{2} \end{aligned}$ .
::气瓶的表面面积为402.1厘米2。

Example 5
::例5

Find the surface area of a cylinder with a diameter of 10 meters and a height of 15 meters.
::找到一个圆柱体的表面面积,直径10米,高度15米。

First, you need to find the radius. Remember that the radius is half the measure of the diameter.
::首先,您需要找到半径。记住半径是直径的半径的一半。

$\begin{aligned} \begin{array}{rcl} r & = & \frac{d}{2} \\ r & = & \frac{10}{2} \\ r & = & 5 \end{array} \end{aligned}$

::= rd2r=102r=5

Next, substitute what you know into the surface area formula.
::接下来,将您所知道的替换为表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π r^{2} + 2 π r h \\ S A & = & 2 π (5)^{2} + 2 π (5) (15) \end{array} \end{aligned}$

::SA=2r2+2rhSA=2(5)2+2(5)(15)

Then, calculate the surface area.
::然后计算表面积

$\begin{aligned} \begin{array}{rcl} S A & = & 2 π (5)^{2} + 2 π (5) (15) \\ S A & = & 2 π (25) + 2 π (75) \\ S A & = & 157.1 + 471.2 \\ S A & = & 628.3 \end{array} \end{aligned}$

::SA=277.1+471.2SA=628.3

The answer is 628.3.
::答案是628.3。

The surface area of the cylinder is $\begin{aligned} 628.3 m^{2} \end{aligned}$ .
::气瓶表面面积为628.3平方米。

Review
::回顾

Use the diagrams to answer the questions below each one. Use 3.14 to approximate $\begin{aligned} π \end{aligned}$ and round your answers to the nearest hundredths place.
::使用图表回答下面的每个问题。使用3. 14 到大约 ° , 并绕过您的答案到最近的一百个地方。

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

2. What is the shape of the base of this figure?
::2. 这个数字的基数是什么形状?

3. How many bases are there?
::3. 有多少个基地?

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

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

6. Which measurement is given the radius or the diameter?
::6. 哪种测量给定半径或直径?

7. What is the surface area of the figure?
::7. 数字的表面积是多少?

Use what you have learned to answer each question.
::用你学到的东西回答每个问题

8. A cylindrical water tank is 35 feet long and it has a radius of 10 feet. How much sheet metal is the tank made of?
::8. 圆柱体水箱长35英尺,半径为10英尺,罐体由多少金属板制成?

9. True or false. You can only find the surface area if you know the volume.
::9. 正确或虚假。只有知道面积,才能找到表面区域。

10. True or false. Surface area and volume measure the same thing.
::10. 真实或虚假,表面面积和体积测量相同。

11. True or false. Surface area measures the outside of a cylinder.
::11. 真实的或假的,表面面积为圆柱体外。

12. True or false. You need the radius or diameter to find the surface area of a cylinder.
::12. 正确或虚假。您需要半径或直径才能找到圆柱体的表面区域。

13. True or false. The radius is one-half of the diameter.
::13. 正确或虚假,半径为直径的半径。

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。

章节大纲

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)

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)