Section: 金字塔量( 深学习) | 8.5 金字塔数量(深学习)-interactive

Section outline

Replica Scale
::复制缩放

When using 3D printers, understanding the effect of scale on is extremely important. Everything that is printed on a 3D printer comes from a 3D file that can be scaled to any size that the printer can support. A scale model can be based on a real-world object or a user-created 3D design. Use the interactive below to see how a 3D printer can change the scale of three-dimensional designs.
::当使用 3D 打印机时, 理解 3D 打印机的 3D 打印机的 3D 比例效果是极为重要的。打印在 3D 打印机上的所有文件都来自 3D 文件, 文件可以缩放到打印机能够支持的任何大小。比例模型可以基于一个真实世界对象或一个用户创建的 3D 设计。使用下面的交互式程序来查看 3D 打印机如何改变三维设计的规模。

INTERACTIVE

3D Printer

You have a 3D printer and you would like to recreate a cube. Instead of making an identical cube you decide to dilate the print to be larger or smaller than the original. Note that the original cube has side length 3 units.
::您有一个 3D 打印机, 您想要重新创建一个立方体。您决定不制作一个相同的立方体, 而是决定将打印放大为大于或小于原始的。请注意, 原始立方体有侧长 3 个单位。

Move the red point to dilate the print.
::移动红点以扩大打印。

Click Print to see the printer create the new cube.
::点击打印以查看打印机创建新立方体。

The solid red cube inside the black printer will be your dilated print.
::黑色打印机内的固态红色立方体将是你的放大打印。

Your device seems to be offline.
Please check your internet connection and try again.

+

Do you want to reset the PLIX?

Yes

No

Progress

0 / 0

Discussion Questions
::讨论问题讨论问题

Setting the scale slider to 1 means that the shape is at its original size. If you set the slider to 2, it doubles the length, width, and height . What does this do to the volume?
::将缩放滑块设置为 1 意味着形状处于其原始大小。如果您将滑块设为 2, 它会使长度、宽度和高度翻倍。这对音量做什么 ?

If you set the slider to 3, it triples the length, width, and height. What does this do to the volume?
::如果您将滑动滑动器设为 3, 它的长度、宽度和高度会增加三倍。这对音量有什么作用 ?

If you increase the scale by a factor of $\begin{aligned} x \end{aligned}$ what do you think will happen to the volume?
::如果将比例乘以 x 的乘数增大, 你会认为音量会发生什么?

If you set the slider to 0.5, it cuts the length, width, and height in half. What does this do to the volume?
::如果您将滑动滑动器设置为 0.5, 它会将长度、宽度和高度减半。这对音量有什么作用 ?

If you decrease the scale by a factor of $\begin{aligned} x \end{aligned}$ what do you think will happen to the volume?
::如果将比例缩小为 x 的乘数, 你会认为音量会发生什么?

Volume of Pyramids
::金字塔体积

I n this section, you will extend your understanding of volume to pyramids . A pyramid is a 3-dimensional solid with a polygonal base and triangular sides meeting at a vertex . This vertex at which the triangular sides meet is known as the apex and is generally the top of the pyramid. A pyramid is defined by the shape of its base. In this section, you will mainly be focused on square pyramids (pyramids with square bases) and rectangular pyramids (pyramids with rectangular bases). When f inding the volume of a pyramid, you are answering the question, “How many cubes can fit inside?” Y ou can use the relationship between a pyramid and a prism with the same base to find the volume. Use the interactive below to explore this relationship.
::在本节中, 您将会将您对体积的理解扩展至金字塔。金字塔是一个三维的固态, 有多边形基和三角边在顶端相会。这个三角边相会的顶点被称为顶点, 通常是金字塔的顶端。金字塔由基座的形状来定义。在这一节中, 您将主要关注方形金字塔( 金字塔有平方基)和矩形金字塔( 有矩形基的金字塔 ) 。当找到金字塔的体积时, 您会回答一个问题 : “ 有多少立方体能适合内面? ” 您可以使用金字塔和同一底基的棱晶之间的关系来查找其体积。使用下面的互动来探索这个关系。

INTERACTIVE

Pyramid vs Prism

Drag the red, blue, and green points to compare the volume of a pyramid to a prism.
::拖曳红色、蓝色和绿色点,以比较金字塔和棱镜的体积。

Drag the slider to change the number of sides of the base.
::拖动滑块以更改基底的边数。

Your device seems to be offline.
Please check your internet connection and try again.

+

Do you want to reset the PLIX?

Yes

No

Discussion Questions
::讨论问题讨论问题

What do you think the relationship is?
::你认为什么关系是什么关系?

Choose any values for the length, width, and height. What do you get when you divide the volume of the rectangular prism by the volume of the pyramid?
::选择长度、宽度和高度的值。当将矩形棱柱的体积除以金字塔的体积时, 您能得到什么 ?

Choose different values for the length, width, and height. What do you get when you divide the volume of the rectangular prism by the volume of the pyramid?
::选择长度、宽度和高度的不同值。当将矩形棱柱的体积除以金字塔的体积时, 您能得到什么 ?

What do you think the relationship is?
::你认为什么关系是什么关系?

If a rectangular prism has a volume of $\begin{aligned} 2, 100 {in}^{3}, \end{aligned}$ what will be the volume of a pyramid with an equal base and equal height?
::如果长方形棱柱体积为2,100英吋3, 金字塔的体积是多少? 金字塔的基数和身高相等。

Using the Formula
::使用公式

This relationship between pyramids and rectangular prisms of equal base and size is true for every type of pyramid. In general, you can use the following formula:
::金字塔和基数和大小相等的矩形棱镜之间的关系对每一种金字塔都是真实的。一般情况下,您可以使用以下公式:

$\begin{aligned} Volume = \frac{Area of the Base \cdot height}{3} \end{aligned}$

::数量=基数区域3

When dealing with square pyramids and rectangular pyramids, the area of the base is length times width. Therefore, you can use a more specific formula:
::当处理平方金字塔和矩形金字塔时,基的面积是长度乘宽。因此,您可以使用一个更具体的公式:

$\begin{aligned} Volume = \frac{length \cdot width \cdot height}{3} \end{aligned}$

::体积=长度

Example
::示例示例示例示例

An obelisk is a solid composed of a pyramid on top of a truncated pyramid. A truncated pyramid is a pyramidal shaped solid that has been cut by a plane parallel to the base. A well-known example of an obelisk is the Washington Monument. The bottom part of the monument is a truncated pyramid, and the top is a square pyramid. The interactive below shows the dimensions , in feet, for the square pyramid on top of the monument. Find the volume of the pyramid.
::方尖碑是一块由金字塔组成的固态方块,它由一个被断裂的金字塔顶部的金字塔组成。一个被断裂的金字塔是一个金字塔形的固态。一个众所周知的方尖碑的例子就是华盛顿纪念碑。纪念碑的底部是一条被断裂的金字塔,顶部是一座方形金字塔。下面的交互面显示了纪念碑顶部的方形金字塔的尺寸(以英尺计)。找到金字塔的体积。

INTERACTIVE

Washington Monument Square Pyramid

Press the button to zoom in and out on the top part of the Washington Monument.
::按下按钮以放大华盛顿纪念碑顶部的进出

When zoomed in, click the blue and green segments to see the dimensions.
::当放大时,单击蓝色和绿色的片段来查看维度。

Your device seems to be offline.
Please check your internet connection and try again.

+

Do you want to reset the PLIX?

Yes

No

The steps you will use are the same as the steps used to find the .
::您要使用的步数与查找步数所用的步数相同。

1. Write the formula.
::1. 编写公式。

$\begin{aligned} Volume = \frac{length \cdot width \cdot height}{3} \end{aligned}$

::体积=长度

2. Substitute the dimensions of the shape into the formula.
::2. 以公式取代形状的尺寸。

$\begin{aligned} Volume = \frac{34.5 \cdot 34.5 \cdot 50}{3} \end{aligned}$

::卷号=34.534.5503

3. Use order of operations to simplify the formula.
::3. 使用操作顺序简化公式。

$\begin{aligned} Volume & = \frac{34.5 \cdot 34.5 \cdot 50}{3} \\ Volume & = \frac{1, 190.25 \cdot 50}{3} \\ Volume & = \frac{59, 512.5}{3} \\ Volume & = 19, 837.5 \end{aligned}$

::卷=1,190.25503 卷=59,51.253 卷=19,837.5

The volume of the pyramid is 19,837.5 ft ³ .
::金字塔的体积是19,837.5平方英尺。

Replica Scale Continued
::继续复制缩放

Example
::示例示例示例示例

The Pyramid of the Sun is an Aztec pyramid in Teotihuacán, Mexico that was constructed around 100 AD. It is currently the seventh largest pyramid in the world, having a volume of 1,265,625 m ³ . The Pyramid of the Sun is a square pyramid with a length and width of 225 meters and a height of 75 meters.
::太阳金字塔是墨西哥Teotihuacán的Aztec金字塔,建造约100个AD。目前,它是世界上第七大金字塔,体积为1,265,625米。太阳金字塔是一个方形金字塔,长宽225米,高75米。

Noreen is looking to 3D print a replica of the pyramid for a school project at a scale of $\begin{aligned} \frac{1}{750} \end{aligned}$ of the real pyramid. What would the dimensions and volume of her replica be?
::诺琳正在寻找 3D 打印一个金字塔的复制品用于一个规模为 1750 的真正的金字塔的学校项目。她的复制品的尺寸和数量会是多少?

A scale of $\begin{aligned} \frac{1}{750} \end{aligned}$ means that you can multiply the dimensions by $\begin{aligned} \frac{1}{750} \end{aligned}$ to find the dimensions of the replica. Multiplying by $\begin{aligned} \frac{1}{750} \end{aligned}$ is equivalent to dividing by 750.
::比例尺为 1750 , 意味着您可以将尺寸乘以 1750 以找到复制的尺寸。乘以 1750 等于以 750 除以 750 。

$\begin{aligned} Length = 225 \cdot \frac{1}{750} = 225 \div 750 = 0.3 meters \\ Width = 225 \cdot \frac{1}{750} = 225 \div 750 = 0.3 meters \\ Height = 75 \cdot \frac{1}{750} = 75 \div 750 = 0.1 meters \end{aligned}$

::长度=2251750=225750=0.3米Width=2251750=2251750=225750=225750=0.3米H8=751750=75750=0.1米

You can now use the formula with the dimensions of the scale replica.
::现在您可以使用带有比例尺复制尺寸的公式。

$\begin{aligned} Volume & = \frac{length \cdot width \cdot height}{3} \\ Volume & = \frac{0.3 \cdot 0.3 \cdot 0.1}{3} \\ Volume & = \frac{0.09 \cdot 0.1}{3} \\ Volume & = \frac{0.009}{3} \\ Volume & = 0.003 m^{3} \end{aligned}$

::卷=0.30.30.13 卷=0.090.13 卷=0.0093 卷=0.003立方米

Use the interactive below to 3D print the pyramid using different scales.
::使用下方的交互效果打印 3D 金字塔,使用不同比例。

INTERACTIVE

Replica Scale continued

The original pyramid is 225 m by 75 m, and this printer scales it down.
::原始金字塔是225米乘75米, 这个打印机缩放下来。

Move the slider to change the scale and press Print to see the pyramid be created.
::移动滑动器以更改缩放, 按打印键以查看金字塔的创建。

Your device seems to be offline.
Please check your internet connection and try again.

+

Do you want to reset the PLIX?

Yes

No

Discussion Questions
::讨论问题讨论问题

What would the volume be in centimeters? There are 100 centimeters in a meter.
::体积是多少厘米?

Why can’t you multiply your answer by 100 to convert it to centimeters?
::为何不能将答案乘以100来转换成厘米?

Could we have divided the real volume of 1,265,625 m³ by 750 to find the answer?
::我们能否将1,265,625立方米的实量除以750来找到答案?

Remember this!
::记住这个!

The volume of a pyramid is one-third the volume of a prism with the same base and height, or $\begin{aligned} V = \frac{B \cdot h}{3} . \end{aligned}$
::金字塔的体积是基数和身高相同的棱晶体体积的三分之一,或V=Bh3。

The volume of rectangular and square pyramids can be found using the formula $\begin{aligned} V = \frac{l \cdot w \cdot h}{3} . \end{aligned}$
::矩形金字塔和平方金字塔的体积可使用公式V=lwh3找到。

If an object is scaled by some factor, its volume will be scaled by that factor three times.
::如果一个对象按某种因数缩放,其体积将按该因数缩放三次。

Section outline

Replica Scale ::复制缩放

Volume of Pyramids ::金字塔体积

Using the Formula ::使用公式

Replica Scale Continued ::继续复制缩放

Remember this! ::记住这个!

Replica Scale
::复制缩放

Volume of Pyramids
::金字塔体积

Using the Formula
::使用公式

Replica Scale Continued
::继续复制缩放

Remember this!
::记住这个!