表面面积 - 网、定义、公式、互动和实例

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What is a Net?
::什么是网络?

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Here are some steps to determine whether a net forms a specific solid:
::以下为确定净额是否构成具体固体的一些步骤:

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1. Make sure that the solid and the net have the same number of faces and that the shapes of the faces of the solid match the shapes of the corresponding faces in the net.
::1. 确保固体和蚊帐的面孔数目相同,确保固体面孔的形状与蚊帐中相应面孔的形状相符。

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2. Visualize how the net is to be folded to form the solid and make sure that all the sides fit together properly.
::2. 想象一下如何折叠网以形成固体,并确保所有各方都适当结合。

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Nets are helpful when we need to find the of the solids. In the image above, you can see an 'unfolded' triangular prism consists of two triangles and three . The triangles are the bases of the prism and the rectangles are the lateral faces .
::当我们需要找到固态时, 网是有用的。 在以上图像中, 您可以看到一个“ 未覆盖的”三角棱镜由两个三角形和三个三角形组成。 三角形是棱镜的底部, 矩形是侧面 。

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Surface Area of a Solid
::固体表面面积

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The surface area of a 3-dimensional object is the measure of the total area of all its faces. This means that one way to find the surface area of a solid is to find the area of its net.
::三维天体的表面区域是其面部总面积的测量值,这意味着找到固体表面区域的一种方法就是找到其网的面积。

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$\begin{align*}\text{Surface area of rectangular prism} = 2ab + 2ac + 2bc = 2(ab + ac + bc)\end{align*}$
::矩形棱晶表面面积=2ab+2ac+2bc=2(ab+ac+bc)

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The surface area of a rectangular prism is the area of the six rectangles that cover it , however y ou don't actually have to figure out all six because  you know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.
::矩形棱柱的表面区域是覆盖它的六个矩形的面积, 然而您实际上不必 找出全部六个矩形, 因为您知道上下是相同的, 前后是相同的, 左侧和右侧是相同的。

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Nets of Solids
::固体净值

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Explore the nets and see how the 2d shape transforms into a 3d shape.
::探索网络,看看2D形状是如何变成3D形状的。

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Rectanglular Prism
::矩形棱柱

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Note that there could be a couple different interpretations of any net. Most prisms have multiple nets.
::请注意,对任何网络可能有几种不同的解释,大多数棱镜都有多个网络。

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CK-12 Interactive: Surface Area and Nets
::CK-12 互动:海面面积和网

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CK-12 Interactive: Surface Area and Nets
::CK-12 互动:海面面积和网

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Find the approximate surface area of the triangular prism below. The base is an .
::查找下方三角棱镜的近似表面区域。 底部是 。

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The net is made of three congruent rectangles and two congruent equilateral triangles .
::网由三个相似矩形和两个相似的等边三角形组成。

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The surface area is the sum of the areas of the five shapes . To find the area of the triangle , you need to know the height of the triangle. Using the Pythagorean Theorem , you can determine that the height is approximately 3.46 inches.
::表面区域是五个形状区域的总和。要找到三角形区域,您需要知道三角形的高度。使用 Pytagoren 理论,您可以确定高度约为 3.46 英寸。

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$\begin{align*}H^2 & = P^2 + B^2 \\ 4^2 & = P^2 + 2^2 \\ P^2 & = 16 - 4 = 12 \\ P & = 3.46 \ \text{inches}\\\end{align*}$
::H2=P2+B242=P2+22P2=16-4=12P=3.46英寸

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Draw the net of a square pyramid .
::绘制平方金字塔的网。

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A square pyramid has a height of 20 inches . Each side of the square base is 12 inches . What is the surface area of the pyramid?
::方形金字塔的高度为20英寸。 方形基底的每侧为12英寸。 金字塔的面积是多少?

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In order to find the surface area, you will need to determine the area of the triangle faces. In order to find the area of each triangle face, you will need the base and height of the triangle. In order to do this, imagine a right triangle standing upright in the pyramid.
::要找到表面积, 您需要确定三角面的面积。 要找到三角面的面积, 您需要三角面的底部和高度。 要做到这一点, 请想象金字塔上右直立的三角形 。

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The height of this triangle is 20 inches and the base of the triangle is 6 inches. It's hypotenuse , which is the height of the triangle face, can be determined with the Pythagorean Theorem.
::此三角形的高度为 20 英寸, 三角形的底部为 6 英寸。 下限是 三角形面的高度, 可以通过 Pytagoren 理论来确定 。

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Now you can find the area of each of the five shapes that make up the net in order to find the surface area.
::现在,你可以找到构成网络的五个形状中的每一个形状的区域,以便找到表面区域。

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$\begin{align*}H^2 & = P^2 + B^2 \\ H^2 & = 20^2 + 6^2 \\ H^2 & = 400 + 36 \\ H & = \sqrt{436} = 20.88 \ \text{inches}\end{align*}$
::H2=P2+B2H2=202+62H2=400+36H=436=20.88英寸

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Identify the solid whose net is given below, based on the base shape .
::根据基准形状,标明其净值如下的固体。

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The solid is .
::固态是。

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CK-12 PLIX: Cross-Sections and Nets
::CK-12 PLIX:跨科和网

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Surface Area of Solids - Examples
::固体表面面积 -- -- 实例

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Example 1
::例1

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A cup of paint covers about 22 square feet. You need to paint all faces of a cube to use as a prop in a play. Each edge of the cube is 2.5 feet long. How much paint will you need to buy?
::一杯咖啡的油漆覆盖约22平方英尺。 您需要油漆立方体的所有面孔才能在剧中用作道具。 立方体的每个边缘都长2.5英尺。 您需要购买多少油漆 ?

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One cup of paint covers about 22 square feet. Each edge of the cube is 2.5 feet long. In order to figure out how much paint you will need, you should find the surface area of the cube.
::一杯咖啡的油漆覆盖约22平方英尺。 立方体的每个边缘长2.5英尺。 为了弄清楚你需要多少油漆, 您应该找到立方体的表面区域 。

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The cube has six congruent square faces . The area of each square is $\begin{align*}(2.5)^2=6.25 \ \text{ft}^2\end{align*}$ . The total surface area of the cube is $\begin{align*}37.5 \ \text{ft}^2\end{align*}$ . You will need $\begin{align*}\frac{37.5}{22}\approx 1.7\end{align*}$ cups of paint.
::立方体有六张正方形正方形。 每个方形的面积是(2.5)2=6.25平方英尺。 立方体的总面积是37.5平方英尺。 您需要37.5221.7杯油漆。

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Example 2
::例2

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Draw the net for a pentagonal prism.
::为五角形棱镜绘制网。

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Example 3
::例3

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The base of a pentagonal prism is the following pentagon :
::五角棱镜的基底是以下五角形:

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Find the area of the pentagon by dissecting the pentagon into five triangles and finding the area of each triangle.
::通过将五角形分解成五个三角形,并找到每个三角形的区域,找到五角形的区域。

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Use the Pythagorean Theorem to find the height of each triangle.
::使用 Pytagorean 理论来查找每个三角形的高度 。

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$\begin{align*}H^2 & = P^2 + B^2 \\ {8.5}^2 & = P^2 + 5^2 \\ P^2 & = 72.25 - 25 = 47.25 \\ P & = \sqrt{47.25} = 6.87 \ \text{inches}\end{align*}$
::H2=P2+P2+B28.52=P2+52P2=72.25-25=47.25P=47.25P=47.25=6.87英寸

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Example 4
::例4

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Find the surface area for a pentagonal prism with a height of 25 inches and a base given in Example 3.
::查找五角棱镜的表面区域,其高度为25英寸,且以例3中给出的基数为基数。

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The pentagonal prism is made of five rectangular faces and two pentagonal faces . Each rectangle is 10 inches by 25 inches.
::五角棱镜由五张长方形面和两张五角形面组成,每个矩形10英寸乘25英寸。

Summary

播放段落 A net is a 2-dimensional shape that can be folded to form a 3-dimensional shape or solid, showing each face and edge of the figure in 2-dimensions. ::网是一个二维形状,可以折叠成三维形状或固态,以二维尺寸显示图的每个面部和边缘。 播放段落 Nets are helpful for finding the surface area of solids, which is the measure of the total area of all its faces. ::蚊帐有助于找到固体表面面积,这是测量其面部总面积的尺度。 播放段落 The surface area of a rectangular prism can be found by calculating the area of its six rectangles, but only three unique rectangles need to be calculated since opposite sides are the same. ::矩形棱柱的表面面积可以通过计算其六个矩形的面积找到,但只需要计算三个独特的矩形面积,因为对立面是相同的。 播放段落 Most prisms have multiple nets, and there could be different interpretations of any net. ::大多数棱镜都有多个网,对任何网可能有不同的解释。

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Surface Area of Solids - Review Questions
::固体表面面积 -- -- 审查问题

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1. Explain the connection between the surface area of a solid and the net of a solid.
::1. 解释固体表面面积与固体网之间的关联。

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2. When stating a surface area, why do you use square units such as " $\begin{align*}in^2\end{align*}$ "?
::2. 在说明表面积时,为什么使用 " in2 " 等平方单位?

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A triangular pyramid has four congruent equilateral triangle faces. Each edge of the pyramid is 6 inches.
::三角金字塔有四个相似的等边三角面,金字塔的边缘各6英寸。

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3. Draw a net for the pyramid.
::3. 绘制金字塔的网。

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4. Find the area of one triangle face.
::4. 找到一个三角面的面积。

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5. Find the surface area of the pyramid.
::5. 寻找金字塔的表面区域。

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A 20 inch tall hexagonal pyramid has a regular hexagon base that can be divided into six equilateral triangles with side lengths of 12 inches, as shown below.
::20英寸高的六边形金字塔有一个普通的六边形基数,可分为6个侧长12英寸的等边三角形,如下文所示。

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6. Draw a net for the pyramid.
::6. 绘制金字塔网。

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7. Find the area of the hexagon base.
::7. 寻找六边形基地的区域。

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8. The pyramid has 6 triangular faces. Use the Pythagorean Theorem to help you to find the height of each of these triangles.
::8. 金字塔有6个三角面,使用毕达哥里安理论来帮助你找到每个三角的高度。

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9. Find the total surface area of the pyramid.
::9. 找出金字塔的总面积。

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A square prism is topped with a square pyramid to create the composite solid below.
::一个平方棱柱被一个平方金字塔覆盖 形成下面的复合固体

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10. Draw a net for the solid.
::10. 绘制固体的网。

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11. There are four triangular faces. Use the Pythagorean Theorem to help you find the height of each of these triangles.
::11. 有四个三角形面,使用毕达哥里安理论来帮助您找到这些三角形的高度。

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12. Find the total surface area of the solid.
::12. 查明固体的总面积。

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A solid has the following net.
::固体有以下的网。

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13. What type of solid is this?
::13. 这是哪种固体?

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14. Find the surface area of the solid.
::14. 寻找固体的表面区域。

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15. What would the net of a cylinder look like? Try to make a sketch.
::15. 圆柱形的网形会是什么样子?

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16. What would the net of a cone look like? Try to make a sketch.
::16. 锥形网长什么样?

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17. Graph the points  $\begin{align*}(2,3)\end{align*}$ and $\begin{align*}(12,3)\end{align*}$ . Label them A and B and connect them to form a segment. Draw vertical segments connecting them to the x-axis. Rotate this shape around the x-axis to form a solid. What figure have you made? Explain. Find the surface area and volume of this figure.
::17. 绘制点数(2,3)和(12,3),并标出A和B,将其连接成一个段。绘制垂直段,将其与x轴连接。在 x 轴周围旋转此形状以形成固体。您做了哪些图?请解释。请查找此图的表面面积和体积。

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18. Graph the points $\begin{align*}(2,0)\end{align*}$ and $\begin{align*}(10,6)\end{align*}$ . Label them A and B and connect them to form a segment. Draw a vertical segment from B to the x-axis. Rotate this shape around the x-axis to form a solid. What figure have you made? Explain. Find the surface area and volume of this figure.
::18. 绘制点数(2,0)和(10,6),标出点数A和B,并将它们标出为A和B, 并连接成一个段。 从 B 绘制垂直段到 x 轴。 在 x 轴周围旋转此形状以形成固体。 您做了哪些数字? 请解释 。 查找此图的表面面积和体积 。

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19. Graph the points $\begin{align*}(24,10)\end{align*}$ and $\begin{align*}(12,5)\end{align*}$ . Form a solid of revolution as described above. Find its surface area and volume.
::19. 绘制各点(24.10点和12.5点)的图,形成上述革命的固体,找出其表面面积和体积。

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Review (Answers)
::审查(答复)

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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 。