Course: 8.9 锥体表面面积

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锥锥体表面区域
Robbie is building a base for a cylindrical fan to put in his room. The truncated cone has a larger radius of 3 feet, a smaller radius of 2 feet, and a slant height of 3 ft. He is going to paint the stand and needs to find the of the base to figure out how much paint to buy. How can Robbie figure out the surface area?
::Robbie正在建立一个基础,让圆柱形扇子放入他的房间。短锥体的半径大为3英尺,小半径小为2英尺,斜坡高为3英尺。他将画一个摊位, 并且需要找到底座来弄清楚要买多少油漆。 Robbie如何找到表面面积?

In this concept, you will learn to find the surface area cones and truncated cones.
::在这个概念中,你会学会找到表层的圆锥和短短的圆锥。

Surface Area
::地表地区

Cones are three-dimensional figures that have a circular base and a point at the top. You can calculate the surface area of a cone using a or a formula. Surface area is the total of the areas of each face in a .
::锥体是三维的图形,其圆形基数和顶部的点数。您可以使用公式或公式计算锥体的表面区域。表面区域是每个面孔在一面的总面积。

Here is what the net of a cone would look like.
::这就是锥形网的长相。

The shaded circle is the base. Remember, cones always have circular bases. The un-shaded portion of the cone represents its side. Technically you don’t call this a face because it has a round edge .
::阴影圆是底盘。记住, 锥形总是有圆形的基座。锥形的未遮盖部分代表其侧面。技术上说,你不会将此称为面孔,因为它有一个圆边。

To find the surface area of a cone, you need to calculate the area of the circular base and the side and add them together. The formula for finding the area of a circle is $\begin{align*}A = \pi r^2\end{align*}$ , where $\begin{align*}r\end{align*}$ is the radius of the circle. You use this formula to find the area of the circular base.
::要找到圆锥形的表面区域, 您需要计算圆锥形基座和侧面的区域, 并把它们加在一起。查找圆锥形区域的公式是 Ar2, 其中 r 是圆形的半径。您使用此公式来查找圆圆基的面积。

To find the area of the cone’s side, you multiply the radius ( $\begin{align*}r\end{align*}$ ), the slant height ( $\begin{align*}s\end{align*}$ ), and $\begin{align*}\pi\end{align*}$ .
::要找到锥形侧的面积,您要乘以半径(r)、倾斜高度(s)和。

$\begin{align*}A = \pi r s\end{align*}$
::阿尔斯

Let’s look at an example.
::让我们举个例子。

Find the surface area of the following cone.
::查找以下锥体的表面区域。

First, calculate the area of the base.
::首先,计算基数的面积。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & π (5)^{2} \\ A & = & 78.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \pi r^2\\ A &=& \pi (5)^2\\ A &=& 78.5 \end{array}\end{align*}$
::2A(5)2A=78.5

Next, calculate the area of the sector , also known as the lateral surface area .
::接下来,计算部门面积,也称为平面表面积。

$\begin{aligned} \begin{array}{rcl} A & = & π (5) (11.7) \\ A & = & 183.8 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \pi (5)(11.7)\\ A &=& 183.8 \end{array}\end{align*}$
::A(5)(11.7)A=183.8

Then, add the area of the base to the lateral surface area.
::然后,在横向表面区域中加上基座区域。

$\begin{aligned} \begin{array}{rcl} A & = & base + lateral Surface \\ A & = & 78.5 + 183.8 \\ A & = & 262.3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \text{base} + \text{lateral Surface} \\ A &=& 78.5 + 183.8 \\ A &=& 262.3 \end{array}\end{align*}$
::A=基数+边面A=78.5+183.8A=262.3

The answer is 262.3.
::答案是262.3。

The surface area of the cone is $\begin{align*}262.3 \ in^2\end{align*}$ .
::锥体的表面面积为2英寸262.3。

A short cut would be to simply use the formula to find the surface area of a cone. Here is the formula for finding the surface area of a cone: $\begin{align*}SA = \pi r^2 + \pi r s\end{align*}$ .
::捷径是简单地使用公式来找到锥体的表面区域。这是找到锥体的表面区域的公式: SAr2rs。

The first part of the formula, $\begin{align*}\pi r^2\end{align*}$ , is simply the area formula for circles. This represents the base area. The second part, $\begin{align*}\pi r s\end{align*}$ , represents the lateral surface area of the cone’s side. You can simply put the pieces together and solve for the area of both parts at once.
::公式的第一部分, °r2, 仅仅是圆形的面积公式。它代表基区。第二部分, °rs, 代表锥体侧的横向表面区域。您可以将两部分的面积同时拼凑和解决。

A truncated cone is one where the point of the cone is cut off leaving two circular bases and a side face.
::短短的圆锥是一个断开圆锥点的圆锥,留下两个圆圆基和一个侧面。

Below is an image of what a truncated cone looks like.
::下面是被截断的锥形的图象。

Notice that with a truncated cone you will have two different circular bases - a top radius and a bottom one. You will have to find the area of both bases plus the area of the sector to find the surface area.
::请注意, 使用短圆锥形, 您将有两个不同的圆形基座—— 上半径和下半径。您必须找到两个基座的面积, 加上扇区的区域, 才能找到表面区域。

The formula for finding the surface area of a truncated cone is:
::寻找短洞锥体表面面积的公式是:

$\begin{align*}SA = \pi [s (R + r) + R^2 + r^2]\end{align*}$
::SA[s(R+r)+R2+r2]

Notice that $\begin{align*}s\end{align*}$ stands for slant height, the capital $\begin{align*}R\end{align*}$ stands for the larger radius and the lowercase $\begin{align*}r\end{align*}$ stands for the smaller radius.
::注意代表倾斜高度首都R代表大半径较小半径代表小半径

Let’s look at an example.
::让我们举个例子。

What is the surface area of a truncated cone with a slant height of 6 cm, a radius of 8 cm and a radius of 6 cm?
::倾斜高度为6厘米、半径为8厘米、半径为6厘米的短流锥形锥形的表面区域是什么?

First, fill in all you know into the surface area formula.
::首先,填充你所知道的填入表面积公式。

$\begin{aligned} \begin{array}{rcl} S A & = & π [s (R + r) + R^{2} + r^{2}] \\ S A & = & π [6 (8 + 6) + 8^{2} + 6^{2}] \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi [s (R+r) + R^2 + r^2] \\ SA &=& \pi [6(8+6) + 8^2 + 6^2] \end{array}\end{align*}$
::SA[s(R+r)+R2+r2]SA[6(8+6)+82+62]

Next, use algebra to solve for the surface area.
::接下来,使用代数解析表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & π [6 (8 + 6) + 8^{2} + 6^{2}] \\ S A & = & π [6 (14) + 64 + 36] \\ S A & = & π [84 + 64 + 36] \\ S A & = & π [184] \\ S A & = & 578.1 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi [6 (8+6) + 8^2 + 6^2] \\ SA &=& \pi [6 (14) + 64 + 36] \\ SA &=& \pi [84 + 64 + 36] \\ SA &=& \pi [184] \\ SA &=& 578.1 \end{array}\end{align*}$
::SA[6(8+6)+82+62]SA[6(14)+64+36]SA[84+64+36]SA[184]SA=578.1

The answer is 578.1.
::答案是578.1。

The surface area of the cone is 578.1 cm ² .
::锥体的表面面积为578.1厘米。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Robbie’s fan stand.
::Robbie的粉丝摊位有问题,

Robbie is building a stand in the shape of a truncated cone. The bottom circular base has a radius of 3 ft. The circular top has a radius of 2 ft. The slant height for the stand is 3 ft. Robbie has to find the surface area so that he knows how much paint he needs.
::Robbie正在建造一个站台, 其形状是短锥形。底圆基的半径为 3 英尺。圆顶的半径为 2 英尺。该摊位的斜坡高度为 3 英尺。 Robbie 必须找到表面区域, 才能知道他需要多少油漆。

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。

$\begin{aligned} \begin{array}{rcl} S A & = & π [s (R + r) + R^{2} + r^{2}] \\ S A & = & π [3 (3 + 2) + 3^{2} + 2^{2}] \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi [s(R+r) + R^2 + r^2] \\ SA &=& \pi [3 (3+2)+ 3^2 + 2^2] \end{array}\end{align*}$
::SA[s(R+r)+R2+r2]SA[3(3+2)+32+22]

Next, use algebra to solve for the surface area.
::接下来,使用代数解析表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & π [3 (3 + 2) + 3^{2} + 2^{2}] \\ S A & = & π [3 (5) + 9 + 4] \\ S A & = & π [15 + 9 + 4] \\ S A & = & π [28] \\ S A & = & 87.96 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi [3(3+2) + 3^2 + 2^2] \\ SA &=& \pi [3(5) + 9+4] \\ SA &=& \pi [15 + 9 + 4] \\ SA &=& \pi [28] \\ SA &=& 87.96 \end{array}\end{align*}$

The answer is 87.96.
::SA[3(3+2)+32+22]SA[3(5)+9+4]SA[15+9+4]SA[28]SA=87.96 答案是87.96.

Robbie needs enough paint to cover 88 ft ² .
::罗比需要足够的油漆来覆盖88平方英尺

Example 2
::例2

Trey is decorating conical party hats for his party by wrapping them in colored tissue paper. Each hat has a radius of 4.2 centimeters and a slant height of 8.6 centimeters. If he wants to wrap six party hats, how much paper will he need?
::Trey用彩色组织纸包着彩色组织纸来装饰党帽。每顶帽子半径4.2厘米,斜高8.6厘米。如果他想要包着6顶党帽,他需要多少纸?

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。

$\begin{aligned} \begin{array}{rcl} S A & = & π r^{2} + π r s \\ S A & = & π (4.2)^{2} + π (4.2) (8.6) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi r^2 + \pi r s \\ SA &=& \pi (4.2)^2 + \pi (4.2)(8.6) \end{array}\end{align*}$
::2rsSA(4.2)2(4.2)(8.6)

Next, use algebra to solve for the surface area.
::接下来,使用代数解析表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & π (4.2)^{2} + π (4.2) (8.6) \\ S A & = & π (17.64) + π (36.12) \\ S A & = & 55.4 + 113.5 \\ S A & = & 168.9 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi (4.2)^2 + \pi (4.2)(8.6) \\ SA &=& \pi (17.64) + \pi (36.12) \\ SA &=& 55.4 + 113.5 \\ SA &=& 168.9 \end{array}\end{align*}$
::SA(17.64)(36.12)SA=55.4+113.5SA=168.9

Then, find the total area of the 6 hats.
::然后找到6顶帽子的总面积

$\begin{aligned} \begin{array}{rcl} S A_{total} & = & 6 \times S A_{one hat} \\ S A_{total} & = & 6 \times 168.9 \\ S A_{total} & = & 1013.4 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA_{\text{total}} &=& 6 \times SA_{\text{one hat}} \\ SA_{\text{total}} &=& 6 \times 168.9 \\ SA_{\text{total}} &=& 1013.4 \end{array}\end{align*}$

The answer is 1013.4.
::SA总额=6xSAone ANSA总额=6x168.9SA总额=101.34 答案是101.34。

Trey would need 1013.4 cm ² of paper to make the six hats.
::Trey需要101.34厘米的纸张才能制造六顶帽子

Example 3
::例3

Find the surface area of a cone with a radius of 4 inches and a slant height of 6 inches.
::找到圆锥体的表面面积,半径为4英寸,斜坡高度为6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。

$\begin{aligned} \begin{array}{rcl} S A & = & π r^{2} + π r s \\ S A & = & π (4)^{2} + π (4) (6) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi r^2 + \pi r s \\ SA &=& \pi (4)^2 + \pi (4)(6) \end{array}\end{align*}$
::2rsSA(4)2(4)(4)(6)

Next, use algebra to solve for the surface area.
::接下来,使用代数解析表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & π (4)^{2} + π (4) (6) \\ S A & = & π (16) + π (24) \\ S A & = & 50.3 + 75.4 \\ S A & = & 125.7 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi (4)^2 + \pi (4)(6) \\ SA &=& \pi (16) + \pi (24) \\ SA &=& 50.3 + 75.4 \\ SA &=& 125.7 \end{array}\end{align*}$
::* SA(4)2(4)(6)SA(16(24)SA=50.3+75.4SA=125.7)

The answer is 125.7.
::答案是125.7

The surface area of the cone is 125.7 in ² .
::锥体的表面面积为125.7英寸2。

Example 4
::例4

Find the surface area of a cone with a radius of 5 feet and a slant height of 8 feet.
::找到圆锥体的表面区域,半径为5英尺,斜坡高度为8英尺。

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。

$\begin{aligned} \begin{array}{rcl} S A & = & π r^{2} + π r s \\ S A & = & π (5)^{2} + π (5) (8) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi r^2 + \pi r s \\ SA &=& \pi (5)^2 + \pi (5) (8) \end{array}\end{align*}$
::2rsSA(5)2(5)(5)(5)(8)(8)

Next, use algebra to solve for the surface area.
::接下来,使用代数解析表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & π (5)^{2} + π (5) (8) \\ S A & = & π (25) + π (40) \\ S A & = & 78.5 + 125.7 \\ S A & = & 204.2 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi (5)^2 + \pi (5) (8) \\ SA &=& \pi (25) + \pi (40) \\ SA &=& 78.5 + 125.7 \\ SA &=& 204.2 \end{array}\end{align*}$
::SA =78.5+125.7SA=204.2

The answer is 204.2.
::答案是204.2。

The surface area of the cone is 204.2 ft ² .
::锥体的表面面积为204.2平方英尺2。

Example 5
::例5

Find the surface area of a cone with a radius of 3 inches and a slant height of 4.5 inches.
::找到圆锥体的表面面积,半径为3英寸,斜坡高度为4.5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。

$\begin{aligned} \begin{array}{rcl} S A & = & π r^{2} + π r s \\ S A & = & π (3)^{2} + π (3) (4.5) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi r^2 + \pi r s \\ SA &=& \pi (3)^2 + \pi (3) (4.5) \end{array}\end{align*}$
::萨罗尔2尔萨(3)2(3)(3)(5)

Next, use algebra to solve for the surface area.
::接下来,使用代数解析表面积。

$\begin{aligned} \begin{array}{rcl} S A & = & π (3)^{2} + π (3) (4.5) \\ S A & = & π (9) + π (13.5) \\ S A & = & 28.3 + 42.4 \\ S A & = & 70.7 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} SA &=& \pi (3)^2 + \pi (3)(4.5) \\ SA &=& \pi (9) + \pi (13.5) \\ SA &=& 28.3 + 42.4 \\ SA &=& 70.7 \end{array}\end{align*}$
::SA(9)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

The answer is 70.7.
::答案是70.7

The surface area of the cone is 70.7 in ² .
::锥体的表面面积为70.7英寸2。

Review
::回顾

Answer the questions below each of the following diagrams. Use 3.14 to approximate $\begin{align*}\pi\end{align*}$ and round your answers to the nearest hundredths place.
::回答下图下的问题。使用3. 14到约 ° , 并绕过您最接近的百位的答案。

1. What is the name of the figure represented in this net?
::1. 这一净额所列数字的名称是什么?

2. What is the diameter of this figure?
::2. 这一数字的直径是多少?

3. What is the slant height of the figure?
::3. 数字的倾斜高度是多少?

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

7. What is the diameter of the base?
::7. 基地的直径是多少?

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

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

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

11. What is the diameter of the base?
::11. 基地的直径是多少?

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

Find the surface area of each cone.
::找到每个锥体的表面区域。

13. $\begin{align*}r = 4 \ in, sh = 5 \ in\end{align*}$
::13 r= 4 英寸,sh= 5 英寸

14. $\begin{align*}r = 5 \ m, sh = 7 \ m\end{align*}$
::14.r=5米,sh=7米

15. $\begin{align*}r = 3 \ cm, sh = 6 \ cm\end{align*}$
::15 r= 3 cm,sh= 6 cm

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

锥锥体表面区域

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