Course: 11.4 气瓶 | The Way To Learn

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圆柱体
Cylinders
::圆柱体

A cylinder is a solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed.
::一个圆柱体是一个坚固的圆形圆形基地,在平行的平面上。圆圈之间的空间是封闭的。

A cylinder has a radius and a height .
::气瓶有一个半径和一个高度。

A cylinder can also be oblique (slanted) like the one below.
::圆柱体也可以像下面那样斜斜(斜斜)。

Surface Area
::地表地区

is the sum of the area of the faces of a solid. The basic unit of area is the square unit .
::是固体表面面积的总和。基本面积单位为平方单位。

Surface Area of a Right Cylinder: $\begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}$ .
::右圆柱表面面积:SA=2°r2+2°rh。

$\begin{aligned} \underset{⏟}{2 π r^{2}} + \underset{⏟}{2 π r} h \\ area of length \\ both of \\ circles rectangle \end{aligned}$
$\begin{align*}& \ \underbrace{ 2 \pi r^2 } \ \ + \ \ \underbrace{ 2 \pi r} h\\ & \text{area of} \quad \ \ \text{length}\\ & \ \text{both} \qquad \ \ \ \text{of}\\ & \text{circles} \quad \ \ \text{rectangle}\end{align*}$
::2r2+2rh 长长的二次曲线矩形区域

Volume
::量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量

To find the of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit. For cylinders, volume is the area of the circular base times the height.
::要找到任何固体, 您必须知道它占用了多少空间。基本体积单位是立方体单位。对于圆柱体, 体积是圆形基数乘以高度的面积。

Volume of a Cylinder: $\begin{align*}V=\pi r^2 h\end{align*}$ .
::气瓶体积:Vr2h。

If an oblique cylinder has the same base area and height as another cylinder, then it will have the same volume. This is due to Cavalieri’s Principle , which states that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.
::如果一个斜体圆柱体与另一个圆柱体的基面积和高度相同,那么其体积就会相同。这要归功于卡瓦列里尼的原则。该原则规定,如果两个固体在每一层的高度和横截面面积相同,那么它们就会具有相同的体积。

What if you were given a solid three-dimensional figure with congruent enclosed circular bases that are in parallel planes? How could you determine how much two-dimensional and three-dimensional space that figure occupies?
::如果给您一个坚实的三维图,与平行平面上的一致的封闭圆形基体相匹配,你如何确定该数字占据了多少二维和三维空间?

Examples
::实例

Example 1
::例1

If the volume of a cylinder is $\begin{align*}484 \pi \ in^3\end{align*}$ and the height is 4 in, what is the radius?
::如果圆柱体的体积是484英寸3,高度是4英寸,半径是多少?

Solve for $\begin{align*}r\end{align*}$ .
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}解决r.

$\begin{aligned} 484 π & = π r^{2} (4) \\ 121 & = r^{2} \\ 11 i n & = r \end{aligned}$
$\begin{align*}484 \pi &= \pi r^2 (4)\\ 121 &= r^2\\ 11 in &= r\end{align*}$
::484r2(4)121=r211in=r

Example 2
::例2

The of the base of a cylinder is $\begin{align*}80 \pi \ cm\end{align*}$ and the height is 36 cm. Find the total surface area.
::气瓶底部为80厘米,高度为36厘米。

We need to solve for the radius, using the circumference.
::我们需要用环绕解决半径的问题

$\begin{aligned} 2 π r & = 80 π \\ r & = 40 \end{aligned}$
$\begin{align*}2 \pi r &=80 \pi\\ r&=40\end{align*}$
::2°r=80°r=40

Now, we can find the surface area.
::现在我们可以找到地表面积了

$\begin{aligned} S A & = 2 π (40)^{2} + (80 π) (36) \\ = 3200 π + 2880 π \\ = 6080 π u n i t s^{2} \end{aligned}$
$\begin{align*}SA &=2 \pi (40)^2+(80 \pi )(36)\\ &= 3200 \pi +2880 \pi\\ &= 6080 \pi \ units^2\end{align*}$
::SA=2__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Example 3
::例3

Find the surface area of the cylinder.
::找到圆柱体的表面区域。

$\begin{align*}r = 4\end{align*}$ and $\begin{align*}h = 12\end{align*}$ .
::r=4和h=12。

$\begin{aligned} S A & = 2 π (4)^{2} + 2 π (4) (12) \\ = 32 π + 96 π \\ = 128 π u n i t s^{2} \end{aligned}$
$\begin{align*}SA &= 2 \pi (4)^2+2 \pi (4)(12)\\ &= 32 \pi +96 \pi\\ &= 128 \pi \ units^2\end{align*}$
::SA=2(4)2+2(4)(12)=32961282单位。

Example 4
::例4

The circumference of the base of a cylinder is $\begin{align*}16 \pi\end{align*}$ and the height is 21. Find the surface area of the cylinder.
::气瓶底部的环形为16°C,高度为21°C,找到气瓶的表面区域。

We need to solve for the radius, using the circumference.
::我们需要用环绕解决半径的问题

$\begin{aligned} 2 π r & = 16 π \\ r & = 8 \end{aligned}$
$\begin{align*}2 \pi r &=16 \pi\\ r&=8\end{align*}$
::2°r=16°r=8

Now, we can find the surface area.
::现在我们可以找到地表面积了

$\begin{aligned} S A & = 2 π (8)^{2} + (16 π) (21) \\ = 128 π + 336 π \\ = 464 π u n i t s^{2} \end{aligned}$
$\begin{align*}SA &=2 \pi (8)^2+(16 \pi )(21)\\ &= 128 \pi +336 \pi\\ &= 464 \pi \ units^2\end{align*}$
::SA=2______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Example 5
::例5

Find the volume of the cylinder.
::找到气瓶的体积。

If the diameter is 16, then the radius is 8.
::如果直径为16,则半径为8。

$\begin{align*}V=\pi 8^2 (21)=1344 \pi \ units^3\end{align*}$
::V82(21)=134443单位

Review
::回顾

Two cylinders have the same surface area. Do they have the same volume? How do you know?
::两个气瓶的表面面积相同。它们有同样的体积吗? 你怎么知道?

A cylinder has $\begin{align*}r = h\end{align*}$ and the radius is 4 cm. What is the volume?
::圆柱体R=h,半径为4厘米。音量是多少?

A cylinder has a volume of $\begin{align*}486 \pi \ ft.^3\end{align*}$ . If the height is 6 ft., what is the diameter?
::圆柱体的体积为486-ft.3。如果高度是6英尺,直径是多少?

A right cylinder has a 7 cm radius and a height of 18 cm. Find the volume and keep your answer in terms of $\begin{align*}\pi.\end{align*}$
::右圆筒的半径为7厘米,高度为18厘米。查找音量,并保持回答中的 °。

Find the volume of the following solids. Round your answers to the nearest hundredth.
::查找以下固体的体积。将您的答复转至最近的第一百个。

Find the value of $\begin{align*}x\end{align*}$ , given the volume.
::查找 x 值, 以给定音量。

$\begin{align*}V=6144 \pi \ units^3\end{align*}$
::V=61443单位

The area of the base of a cylinder is $\begin{align*}49 \pi \ in^2\end{align*}$ and the height is 6 in. Find the volume.
::圆柱体的底部区域为 49 in2, 高度为 6。请查找音量。

The circumference of the base of a cylinder is $\begin{align*}34 \pi \ cm\end{align*}$ and the height is 20 cm. Find the total surface area.
::圆柱体的底部环形为34厘米,高度为20厘米。找出总表面面积。

The lateral surface area of a cylinder is $\begin{align*}30 \pi \ m^2\end{align*}$ and the height is 5m. What is the radius?
::圆柱体的横向表面面积为 30°°m2,高度为 5 m。半径是多少?

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。

Section outline

General

圆柱体

Cylinders ::圆柱体

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)

Cylinders
::圆柱体

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)