课程： 11.5 金字塔 | The Way To Learn

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金金字石
Pyramids
::金金字石

A pyramid is a solid with one base and lateral faces that meet at a common vertex . The edges between the lateral faces are lateral edges . The edges between the base and the lateral faces are base edges .
::金字塔是一个坚固的金字塔,一个基点和横向面,在共同的顶点相交。横向面之间的边缘是横向边缘。基点和横向面之间的边缘是基点边缘。

A regular pyramid is a pyramid where the base is a regular polygon . All regular pyramids also have a slant height , which is the height of a lateral face . A non-regular pyramid does not have a slant height.
::普通金字塔是金字塔,其底部是普通多边形。所有普通金字塔都有一个斜高,即横向面部的高度。非普通金字塔没有斜高。

Surface Area
::地表地区

is a two-dimensional measurement that is the total area of all surfaces that bound a solid. The basic unit of area is the square unit . For pyramids, we will need to use the slant height, which is labeled $\begin{align*}l\end{align*}$ , to find the area of each triangular face.
::是一个二维的测量数据,是所有将固体捆绑在一起的表面的总面积。基本区域单位是平方单位。对于金字塔,我们需要使用标有I的斜坡高度来找到每个三角形的面积。

Surface Area of a Regular Pyramid: If $\begin{align*}B\end{align*}$ is the area of the base, and $\begin{align*}n\end{align*}$ is the number of triangles, then $\begin{align*}SA=B+\frac{1}{2} nbl\end{align*}$ .
::普通金字塔的表面区域:如果B是基底区域,n是三角数,那么SA=B+12nbl。

The shows the surface area of a pyramid. If you ever forget the formula, use the net.
::显示金字塔的表面区域。如果您忘记公式, 请使用网。

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.
::要找到任何固体,你必须弄清楚它占用了多少空间。基本体积单位是立方体单位。

Volume of a Pyramid: $\begin{align*}V=\frac{1}{3} Bh\end{align*}$ where $\begin{align*}B\end{align*}$ is the area of the base.
::金字塔体积:V=13Bh,B是基地区域。

What if you were given a solid three-dimensional figure with one base and lateral faces that meet at a common vertex? How could you determine how much two-dimensional and three-dimensional space that figure occupies?
::如果给了你一个坚固的三维图,一个基面和面部相邻的平面,在一个共同的顶点相遇?你如何确定该图占据了多少二维和三维空间?

Examples
::实例

Example 1
::例1

Find the slant height of the square pyramid.
::找到方形金字塔的倾斜高度

The slant height is the hypotenuse of the right triangle formed by the height and half the base length. Use the Pythagorean Theorem .
::倾斜高度是右三角形的下限,右三角形由高度和基长的一半组成。使用 Pytagoren 理论。

$\begin{aligned} 8^{2} + 24^{2} & = l^{2} \\ 640 & = l^{2} \\ l = \sqrt{640} & = 8 \sqrt{10} \end{aligned}$
$\begin{align*}8^2+24^2 &= l^2\\ 640 &= l^2\\ l = \sqrt{640} & = 8\sqrt{10}\end{align*}$
::82+242=l2640=l2l=640=810

Example 2
::例2

Find the surface area of the regular triangular pyramid.
::寻找普通三角金字塔的表面区域。

“Regular” tells us the base is an . Let’s draw it and find its area.
::“Regular”告诉我们基地是一个基地。

$\begin{align*}B=\frac{1}{2} \cdot 8 \cdot 4\sqrt{3}=16\sqrt{3}\end{align*}$
::B=12843=163

The surface area is:
::表面面积为:

$\begin{align*}SA=16\sqrt{3}+\frac{1}{2} \cdot 3 \cdot 8 \cdot 18=16\sqrt{3}+216 \approx 243.71\end{align*}$
::SA=163+123818=163+216243.71

Example 3
::例3

If the lateral surface area of a regular square pyramid is $\begin{align*}72 \ ft^2\end{align*}$ and the base edge is equal to the slant height. What is the length of the base edge?
::如果正方形金字塔的平面表面面积为72平方英尺,基底边缘等于倾斜高度。基底边缘的长度是多少?

In the formula for surface area, the lateral surface area is $\begin{align*}\frac{1}{2} nbl\end{align*}$ . We know that $\begin{align*}n = 4\end{align*}$ and $\begin{align*}b = l\end{align*}$ . Let’s solve for $\begin{align*}b\end{align*}$ .
::在表面积公式中,平面表面积为12nbl。我们知道 n=4和b=l。让我们解决b。

$\begin{aligned} \frac{1}{2} n b l & = 72 f t^{2} \\ \frac{1}{2} (4) b^{2} & = 72 \\ 2 b^{2} & = 72 \\ b^{2} & = 36 \\ b & = 6 f e e t \end{aligned}$
$\begin{align*}\frac{1}{2} nbl &= 72 \ ft^2\\ \frac{1}{2} (4) b^2 &= 72\\ 2b^2 &= 72\\ b^2 &= 36\\ b &= 6 feet\end{align*}$
::12nbl=72 ft212(4)b2=722b2=72b2=36b=6fet

Example 4
::例4

Find the height and then volume of the pyramid.
::找到金字塔的高度和体积

In this example, we are given the slant height. Use the Pythagorean Theorem.
::在这个例子中,我们被赋予倾斜高度。使用毕达哥伦神话。

$\begin{aligned} 7^{2} + h^{2} & = 25^{2} \\ h^{2} & = 576 \\ h & = 24 \end{aligned}$
$\begin{align*}7^2+h^2 &=25^2\\ h^2 &= 576\\ h &= 24\end{align*}$
::72+h2=252h2=576h=24

$\begin{align*}V=\frac{1}{3} (14^2)(24)=1568 \ units^3\end{align*}$
::V=13(142)(24)=1568单位3

Example 5
::例5

Find the volume of the pyramid with a right triangle as its base.
::找到金字塔的体积,以右三角为基点。

The base of the pyramid is a right triangle. The area of the base is $\begin{align*}\frac{1}{2} (14)(8)=56 \ units^2\end{align*}$ .
::金字塔的底部是一个右三角形。基座的面积是12(14)(8)=562单位。

$\begin{align*}V=\frac{1}{3} (56)(17) \approx 317.33 \ units^3\end{align*}$
::V=13(56)(17) 317.33单位3

Example 6
::例6

A rectangular pyramid has a base area of $\begin{align*}56 \ cm^2\end{align*}$ and a volume of $\begin{align*}224 \ cm^3\end{align*}$ . What is the height of the pyramid?
::长方形金字塔的基本面积为56厘米2,体积为224厘米3。金字塔的高度是多少?

Use the formula for volume and plug in the information we were given. Then solve for the height.
::使用音量和插件的公式, 并插入我们所得到的信息。然后解析高度。

$\begin{aligned} V & = \frac{1}{3} B h \\ 224 & = \frac{1}{3} \cdot 56 h \\ 12 & = h \end{aligned}$
$\begin{align*}V &= \frac{1}{3} Bh\\ 224 &= \frac{1}{3} \cdot 56h\\ 12 &= h\end{align*}$
::V=13Bh224=13-56h12=h

Review
::回顾

Fill in the blanks about the diagram to the left.
::填入左边图表的空白。

$\begin{align*}x\end{align*}$ is the ___________.
::x 是。

The slant height is ________.
::倾斜高度是______________________________________________________________________________________________________________________________________

$\begin{align*}y\end{align*}$ is the ___________.
::y是... 。

The height is ________.
::身高是___________________________________________________________________________________________________________________________________________

The base is _______.
::基数是_________________________________________________________________________________________________________________________________________________________________________

The base edge is ________.
::底边是________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

For questions 7-8, sketch each of the following solids and answer the question. Your drawings should be to scale, but not one-to-one. Leave your answer in simplest radical form.
::对于问题7-8, 请绘制以下每块固体的草图并回答问题。您的图纸应该是比例尺的, 而不是一对一。请将答案保留在最简单的激进形式。

Draw a square pyramid with a base length of 18 in and a height of 12 in. What is the slant height of the pyramid?
::绘制平方形金字塔,底长18英寸,高12英寸。金字塔的斜高是多少?

Draw a regular tetrahedron (a triangular pyramid with equilateral faces). If the edge length is 6 what is the height of the pyramid?
::绘制正则四面形( 带有等边面的三角金字塔) 。如果边缘长度是 6, 金字塔的高度是 6 ?

Find the slant height, $\begin{align*}l\end{align*}$ , of one lateral face in each pyramid. Round your answer to the nearest hundredth.
::在每个金字塔上找到一个横向面孔的倾斜高度, 1。将您的答案绕到最近的一百位。

Find the surface area and volume of the regular pyramid. Round your answers to the nearest hundredth. (The Median Theorem will be helpful in finding the height of the triangular pyramids.)
::查找正则金字塔的表面面积和体积。将您的答案绕到最接近的第一百个。 (中位元定理将有助于找到三角金字塔的高度。)

A regular tetrahedron has four equilateral triangles as its faces. Keep square roots for the following questions to find exact answers.

Find the height of one of the faces if the edge length is 6 units.
::如果边缘长度为 6 单位, 则查找面部的高度。

Find the area of one face.
::找到一张脸的面积

Find the total surface area of the regular tetrahedron.
::查找正则四面体的总表面面积。

::普通四面形的正四面形有4个等边三角形。保持以下问题的正方根以找到准确的答案。如果边缘长度为 6 个单位, 则查找其中一张脸的高度。查找一张脸的面积。查找普通四面形的总表面面积。

If the surface area of a square pyramid is $\begin{align*}40 \ ft^2\end{align*}$ and the base edge is 4 ft, what is the slant height?
::如果平方金字塔的表面面积是40平方英尺,基边是4平方英尺,斜坡高度是多少?

If the lateral area of a square pyramid is $\begin{align*}800 \ in^2\end{align*}$ and the slant height is 16 in, what is the length of the base edge?
::如果平方金字塔的平面面积是800英寸2, 倾斜高度是16英寸, 基边的长度是多少?

If the lateral area of a regular triangle pyramid is $\begin{align*}252 \ in^2\end{align*}$ and the base edge is 8 in, what is the slant height?
::如果一个普通三角金字塔的平面面积是252英寸2, 底边是8英寸, 斜坡高是多少?

The volume of a square pyramid is 72 square inches and the base edge is 4 inches. What is the height?
::平方金字塔的体积是72平方英寸,基底边缘是4英寸。身高是多少?

The volume of a triangle pyramid is $\begin{align*}170 \ in^3\end{align*}$ and the base area is $\begin{align*}34 \ in^2\end{align*}$ . What is the height of the pyramid?
::三角金字塔的体积是170英寸3 基面积是34英寸2 金字塔的高度是多少?

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。

章节大纲

常规

金金字石

Pyramids ::金金字石

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

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

Pyramids
::金金字石

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

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)