Course: 10.9 锥体表面区域

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 锥锥体表面区域

Collapse Expand
锥锥体表面区域
Darren and Arnold are helping set up teepees for a Native American festival. They want to know how much canvas is required for each teepee. The teepees are cone-shaped, contain canvas flooring, and have no openings with the front flap closed. The radius of the floor of each of the teepees is 6 feet and the slant height is 12 feet. How much canvas is required for each teepee?
::Darren 和 Arnold 正在帮助为美洲原住民节建立排风板。他们想知道每个排风板需要多少帆布。排风板是锥形的, 装有排风板地板, 并且没有前扇关闭的开口。每个排风板的半径是6英尺, 斜坡高度是12英尺。每个排风板需要多少排风板?

In this concept, you will learn how to find the of cones.
::在这个概念中,你会学会如何找到锥形。

Finding the Surface Area of a Cone
::查找一个锥体的表面区域

Cones have different than solids with polygonal faces. Imagine you could unroll a cone.
::锥体与具有多边形面的固体有不同之处。想象一下你可以解开锥体。

The circle is the base. The larger semi-circular portion represents its side (this is not called a face because it has a round edge).
::圆是基数。更大的半圆半圆部分代表它的侧面(这不称为面,因为它有一个圆边)。

To find the surface area of a cone, you 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 and $\begin{align*}\pi\end{align*}$ (pi) is a constant which, when rounded, is equal to 3.14 . This formula is used to find the area of the circular base.
::要找到锥形的表面区域,您要计算圆形基座和侧面的面积,然后将其相加。查找圆形区域的公式是 Ar2, 其中 r 是圆的半径, (pi) 是一个常数, 当四舍五入时, 该常数等于 3. 14。此公式用于查找圆形基座的面积。

The side of the cone is actually a piece of a circle, called a sector . The size of the sector is determined by the ratio of the cone’s slant height to its radius, or $\begin{align*}\frac{s}{r}\end{align*}$ . To find the area of the sector, take the area of the portion of the circle:
::锥形的侧面实际上是圆形的一块块,称为区块。扇形的大小取决于锥体倾斜高度与其半径或弧度之比。要找到扇形的面积,请选择圆形部分的面积:

$\begin{align*}A = \pi r^2 \cdot \frac{s}{r}\end{align*}$ . This simplifies to $\begin{align*}\pi rs\end{align*}$ . To find the area of the cone’s side, multiply the radius by pi , and then multiply the resulting value by the slant height.
::Ar2sr. 此简化到s。要找到锥形侧的区域, 请将半径乘以 pi 乘以 pi, 然后将结果值乘以倾斜高度。

Let's look at an example. What is the surface area of the figure below?
::让我们举个例子。下图的表面面积是多少?

First calculate the area of the bottom face (base) and the side:
::首先计算底部脸部( 底部) 和侧面的区域 :

$\begin{aligned} bottom face & side \\ A = π r^{2} & A = π r s \\ π (5)^{2} & π (5) (11) \\ 25 π & 55 π \\ 25 \times 3.14 & 55 \times 3.14 \\ 78.5 & 172.7 \end{aligned}$
$\begin{align*}& \text{bottom face} && \text{side}\\ & A = \pi r^2 && A = \pi rs\\ & \pi (5)^2 && \pi (5) (11)\\ & 25 \pi && 55 \pi\\ & 25 \times\ 3.14 && 55 \times\ 3.14\\ & 78.5 && 172.7\end{align*}$
::r2Ars(5)2(5)(5)(11) 255525× 3.1455× 3.1478.572.7

Next, add these together to find the surface area of the entire cone, remembering to include the appropriate unit of measurement.
::之后,将这些加在一起,以找到整个锥体的表面区域,记住包括适当的测量单位。

$\begin{aligned} bottom face side surface area \\ 78.5 + 172.7 = 251.2 i n .^{2} \end{aligned}$
$\begin{align*}& \text{bottom face} \qquad \quad \text{side} \qquad \quad \ \text{surface area}\\ & 78.5 \qquad \quad \ \ + \quad \ 172.7 \ \ = \ \ \ 251.2 \ in.^2\end{align*}$
::表面表层78.5+172.7=251.2英寸

Cones have a different surface area formula because they have a circular base. But the general idea is the same as for other solids. One formula can be used as a short cut to combine the area of the circular base and the area of the cone’s side as follows:
::锥体具有不同的表面面积公式,因为它们有一个圆形基体。但一般概念与其他固体相同。一种公式可以作为捷径,将圆形基体的面积和锥体侧的面积结合起来,具体如下:

$\begin{align*}SA = \pi r^2 + \pi rs\end{align*}$
::萨罗尔2

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 represents the area of the cone’s side. 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。第二部分代表锥体的侧面区域。简单地将两部分的碎片组合并同时解决。

Let's look at another example.
::让我们再举一个例子。

What is the surface area of the cone?
::锥形的表面面积是多少?

You can see that the radius of this cone is 3 inches and the slant height is 9 inches.
::你可以看到这个锥体的半径是3英寸倾斜高度是9英寸

First, plug the values for $\begin{align*}r\end{align*}$ and $\begin{align*}s\end{align*}$ into the surface area formula and multiply the values together:
::首先,将 r 和 s 的值插入表面积公式,然后将数值相乘:

$\begin{aligned} S A & = π r^{2} + π r s \\ S A & = π (3^{2}) + π (3) (9) \end{aligned}$
$\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (3^2) + \pi (3)(9)\\\end{align*}$
:32) (3)(9)

$\begin{aligned} S A & = 9 π + 27 π \end{aligned}$
$\begin{align*}SA & = 9 \pi + 27 \pi\\\end{align*}$
::SA=927

Next, add the values together:
::下一步,将数值加在一起:

$\begin{aligned} S A & = 9 π + 27 π \\ S A & = 36 π \end{aligned}$
$\begin{align*}SA & = 9 \pi + 27 \pi\\ SA & = 36 \pi\\\end{align*}$
::SA=927SA=36______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:
::然后,替换 pi 的值, 并乘以答案, 记住包含适当的计量单位 :

$\begin{aligned} S A & = 36 π \\ S A & = 36 \times 3.14 \\ S A & = 113.04 i n .^{2} \end{aligned}$
$\begin{align*}SA & = 36 \pi\\ SA & = 36\ \times\ 3.14\\ SA & = 113.04 \ in.^2\end{align*}$
::SA=36-SA=36 × 3.14SA=113.04英寸。

The answer is this cone has a surface area of 113.04 square inches.
::答案是这个锥体的面积是113.04平方英寸

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Darren and Arnold and their teepee construction.
::早些时候,你被问及 Darren和Arnold 以及他们的轮胎建筑问题。

They need to know how much canvas is required if the radius of each teepee is 6 feet and the slant height is 12 feet.
::他们需要知道需要多少帆布,如果每个轮胎的半径是6英尺,倾斜高度是12英尺。

First, plug the values for $\begin{align*}r\end{align*}$ and $\begin{align*}s\end{align*}$ into the surface area formula and multiply the values:
::首先,将 r 和 s 的值插入表面积公式并乘以数值:

$\begin{aligned} S A & = π r^{2} + π r s \\ S A & = π (6^{2}) + π (6) (12) \end{aligned}$
$\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (6^2) + \pi (6)(12)\\\end{align*}$
$\begin{aligned} S A & = 36 π + 72 π \end{aligned}$
$\begin{align*}SA & = 36 \pi + 72 \pi\\\end{align*}$
:62)(6)(12)SA=3672

Next, add the values together:
$\begin{aligned} S A & = 36 π + 72 π \\ S A & = 108 π \end{aligned}$
$\begin{align*}SA & = 36 \pi + 72 \pi\\ SA & = 108 \pi\\\end{align*}$
::接下来,将数值加在一起:SA=3672SA=108

Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:
$\begin{aligned} S A & = 108 π \\ S A & = 108 \times 3.14 \\ S A & = 339.12 f t .^{2} \end{aligned}$
$\begin{align*}SA & = 108 \pi\\ SA & = 108\ \times\ 3.14\\ SA & = 339.12 \ ft.^2\end{align*}$
::然后,将 pi 的值替换为 pi , 并乘以答案, 记住要包含适当的计量单位: SA=108QSA=108 × 3. 14SA=339.12 ft.2

The answer is the surface area of each teepee is 339.12 square feet. The canvas needed to build one teepee is 339.12 square feet.
::答案是每个轮胎的表面面积为339.12平方英尺,建造一个轮胎所需的帆布为339.12平方英尺。

Example 2
::例2

A cone has a radius of 2.5 meters and a slant height of 7.5 meters. What is its surface area?
::锥体半径为2.5米,斜高度为7.5米。

First, plug the values for $\begin{align*}r\end{align*}$ and $\begin{align*}s\end{align*}$ into the surface area formula and multiply the values together:
::首先,将 r 和 s 的值插入表面积公式,然后将数值相乘:

$\begin{aligned} S A & = π r^{2} + π r s \\ S A & = π ({2.5}^{2}) + π (2.5) (7.5) \end{aligned}$
$\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (2.5^2) + \pi (2.5)(7.5)\\\end{align*}$
:2.5)(7.5)

$\begin{aligned} S A & = 6.25 π + 18.75 π \end{aligned}$
$\begin{align*}SA & = 6.25 \pi + 18.75 \pi\\\end{align*}$
::SA=6.2518.75______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Next, add the values together:
::下一步,将数值加在一起:

$\begin{aligned} S A & = 6.25 π + 18.75 π \\ S A & = 25 π \end{aligned}$
$\begin{align*}SA & = 6.25 \pi + 18.75 \pi\\ SA & = 25 \pi\\\end{align*}$
::SA=6.2518.75_SA=25____________________________________________________________________________________________________________________________________________________________________________

Then, replace the value for pi and multiply, remembering to include the appropriate unit of measurement in the answer:
::然后,替换 pi 和乘数的值, 记住在回答中包括适当的计量单位 :

$\begin{aligned} S A & = 25 π \\ S A & = 25 \times 3.14 \\ S A & = 78.5 i n .^{2} \end{aligned}$
$\begin{align*}SA & = 25 \pi\\ SA & = 25 \times\ 3.14\\ SA & = 78.5 \ in.^2\end{align*}$
::SA=25-SA=25x3.14SA=78.5英寸。

The answer is the cone has a surface area of 78.5 square inches.
::答案是锥体表面面积78.5平方英寸

Example 3
::例3

Find the surface area of a cone with a radius = 4 in and slant height = 6 in.
::查找圆锥体的表面面积,半径=4,斜高度=6。

First, plug the values for $\begin{align*}r\end{align*}$ and $\begin{align*}s\end{align*}$ into the surface area formula and multiply the values together:
::首先,将 r 和 s 的值插入表面积公式,然后将数值相乘:

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

$\begin{aligned} S A & = 16 π + 24 π \end{aligned}$
$\begin{align*}SA & = 16 \pi + 24 \pi\\\end{align*}$
:42) (4)(6) SA=1624

Next, add the values together:
$\begin{aligned} S A & = 16 π + 24 π \\ S A & = 40 π \end{aligned}$
$\begin{align*}SA & = 16 \pi + 24 \pi\\ SA & = 40 \pi\\\end{align*}$
::接下来,将数值加在一起:SA=1624SA=40

Then, replace the value for pi and multiply, remembering to include the appropriate unit of measurement:
::然后,替换 pi 和乘数的值,记住包括适当的计量单位:

$\begin{aligned} S A & = 40 π \\ S A & = 40 \times 3.14 \\ S A & = 125.6 i n .^{2} \end{aligned}$
$\begin{align*}SA & = 40 \pi\\ SA & = 40 \times\ 3.14\\ SA & = 125.6 \ in.^2\end{align*}$
::SA=40-SA=40x3.14SA=125.6英寸

The answer is the surface area of this cone is $\begin{align*}125.6 \ in^2\end{align*}$ .
::答案是这个锥体的表面面积是125.6英寸2。

Example 4
::例4

Find the surface area of a cone with radius = 3.5 in and slant height = 5 in.
::找到圆锥形的表面面积,半径=3.5英寸,斜高度=5英寸。

First, plug the values for $\begin{align*}r\end{align*}$ and $\begin{align*}s\end{align*}$ into the surface area formula and multiply the values:
::首先,将 r 和 s 的值插入表面积公式并乘以数值:

$\begin{aligned} S A & = π r^{2} + π r s \\ S A & = π ({3.5}^{2}) + π (3.5) (5) \end{aligned}$
$\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (3.5^2) + \pi (3.5)(5)\\\end{align*}$

$\begin{aligned} S A & = 12.25 π + 17.5 π \end{aligned}$
$\begin{align*}SA & =12.25 \pi + 17.5 \pi\\\end{align*}$
:3.52)(3.5)(5) SA=12.2517.5

Next, add the values together:
$\begin{aligned} S A & = 12.25 π + 17.5 π \\ S A & = 29.75 π \end{aligned}$
$\begin{align*}SA & = 12.25 \pi + 17.5 \pi\\ SA & = 29.75 \pi\\\end{align*}$
::接下来,将数值加在一起:SA=12.2517.5.SA=29.75

Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:
::然后,替换 pi 的值, 并乘以答案, 记住包含适当的计量单位 :

$\begin{aligned} S A & = 29.75 π \\ S A & = 29.75 \times 3.14 \\ S A & = 93.42 i n .^{2} \end{aligned}$
$\begin{align*}SA & = 29.75 \pi\\ SA & = 29.75\ \times\ 3.14\\ SA & = 93.42 \ in.^2\end{align*}$
::SA=29.75-SA=29.75 × 3.14SA=93.42

The answer is the surface area of this cone is $\begin{align*}93.42 \ in^2\end{align*}$ .
::答案是这个锥体的表面面积是93.42英寸2。

Example 5
::例5

Find the surface area of a cone with a radius = 5 m and slant height = 7 m.
::找到圆锥形的表面面积,半径=5米,斜坡高度=7米。

First, plug the values for $\begin{align*}r\end{align*}$ and $\begin{align*}s\end{align*}$ into the surface area formula and multiply the values:
$\begin{aligned} S A & = π r^{2} + π r s \\ S A & = π (5^{2}) + π (5) (7) \end{aligned}$
$\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (5^2) + \pi (5)(7)\\\end{align*}$

$\begin{aligned} S A & = 25 π + 35 π \end{aligned}$
$\begin{align*}SA & = 25 \pi + 35 \pi\\\end{align*}$ Next, add the values together:
$\begin{aligned} S A & = 25 π + 35 π \\ S A & = 60 π \end{aligned}$
$\begin{align*}SA & = 25 \pi + 35 \pi\\ SA & = 60 \pi\\\end{align*}$ Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:
$\begin{aligned} S A & = 60 π \\ S A & = 60 \times 3.14 \\ S A & = 188.4 m .^{2} \end{aligned}$
$\begin{align*}SA & = 60 \pi\\ SA & = 60\ \times\ 3.14\\ SA & = 188.4 \ m.^2\end{align*}$
::首先,将 r 和 s 的值插入表面积公式, 并乘以以下值: SA @r2rsSA}(52) (5)(7) SA=2535Next, 将数值加在一起: SA=2535SA=60然后, 替换 pi 的值, 并乘以答案, 记住包括适当的测量单位: SA=60SA=60×3. 14SA=188.4 m.2

The answer is the surface area of this cone is $\begin{align*}188.4 \ m^2\end{align*}$ .
::答案是这个锥体的表面面积是188.4平方米。

Review
::回顾

Find the surface area of each cone. Remember that $\begin{align*}sh\end{align*}$ means slant height and $\begin{align*}r\end{align*}$ means radius. Use 3.14 for the value $\begin{align*}\pi.\end{align*}$
::查找每个锥锥体的表面区域。记住, 圆形是指倾斜高度和 r 表示半径。值 = 3. 14 使用 3. 14 。

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

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

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

$\begin{align*}r = 5 \ mm, \ sh = 8 \ mm\end{align*}$
::r=5毫米,sh=8毫米

$\begin{align*}r = 8 \ in, \ sh = 10 \ in\end{align*}$
::r= 8 英寸, sh= 10 英寸

$\begin{align*}r = 11 \ cm, \ sh = 14 \ cm\end{align*}$
::r=11厘米,sh=14厘米

$\begin{align*}r = 12 \ in, \ sh = 16 \ in\end{align*}$
::r=12英寸, sh=16英寸

$\begin{align*}r = 3.5 \ cm, \ sh = 6 \ cm\end{align*}$
::r=3.5厘米,sh=6厘米

$\begin{align*}r = 4.5 \ mm, \ sh = 7 \ mm\end{align*}$
::r=4.5毫米,sh=7毫米

$\begin{align*}r = 6 \ cm, \ sh = 8 \ cm\end{align*}$
::r=6厘米,sh=8厘米

$\begin{align*}r = 7.5 \ cm, \ sh = 9 \ cm\end{align*}$
::r=7.5厘米,sh=9厘米

$\begin{align*}r = 10 \ cm, \ sh = 12 \ cm\end{align*}$
::r=10厘米,sh=12厘米

$\begin{align*}r = 16 \ cm, \ sh = 18 \ cm\end{align*}$
::r=16厘米,sh=18厘米

$\begin{align*}r = 13 \ cm, \ sh = 20 \ cm\end{align*}$
::r=13厘米,sh=20厘米

$\begin{align*}r = 15.5 \ cm, \ sh = 18.5 \ cm\end{align*}$
::r=1.55厘米,sh=18.5厘米

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。

Section outline

General

锥锥体表面区域

Finding the Surface Area of a Cone ::查找一个锥体的表面区域

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)

Finding the Surface Area of a Cone
::查找一个锥体的表面区域

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)