固体的面部、边缘和顶部

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Martha is visiting the famous Louvre, in Paris, France.  She is awed by the fascinating structure outside the museum.  She wants to identify the structure according to its attributes. 
::Martha正在法国巴黎参观著名的卢浮宫,她被博物馆外的迷人结构所迷惑。她想根据博物馆的属性来辨别结构。

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What is the name of this and how many faces does it have?
::这个名字叫什么 有多少张脸?

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In this concept, you will learn how to identify a figure according to faces, edges and vertices , as well as by the shape of the base.
::在这个概念中,你将学会如何根据面孔、边缘和脊椎以及基体的形状来识别一个数字。

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Using Faces, Edges, and Vertices to Identify a Solid
::使用面、边和顶点识别固体

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Every solid figure, with the exception of a sphere , has one or more faces , or flat surfaces.
::除球体外,每个坚固的体形都有一个或多个面孔或平面。

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Let's look at an example.
::让我们举个例子。

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The figure above has a face on the bottom, called the base , and another on the top. It has four faces around the sides. Therefore, it has six faces in all.
::上图的底部有一张脸,叫做底部,上图的顶部有一张脸,两侧有四张脸。因此,上图的底部有六张脸。

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The shapes of the faces are . Since the shape on the bottom and the top are the same, the general category of this shape is called a prism .  Prisms are more specifically named by the shape of the base.  In this case, the shape of the base is a rectangle ; so this solid figure is called a rectangular prism .
::面部的形状是 。 由于底部和顶部的形状是相同的, 此形状的一般类别被称为棱镜。 棱柱由底部的形状更具体地命名。 在这种情况下, 底部的形状是一个矩形; 因此, 此固态图被称为矩形棱柱 。

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Let's look at an example.
::让我们举个例子。

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This figure has four faces. The base, or the face on the bottom, is a triangle , and the sides are triangles that meet at a vertex at the top. The general category for this solid figure is a pyramid . Notice that the base of the pyramid is a triangle. The specific name of this soid figure is called a triangular pyramid . As with the prism, the base indicates the specific name of the figure.
::此图有四张面孔。 底部的底部或面孔是一个三角形, 侧面是三角形, 在顶部的顶点相交。 此固态图的普通类别是金字塔。 请注意, 金字塔的底部是三角形。 此粘土图的具体名称被称为三角形。 与棱镜一样, 底部是该图的具体名称 。

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You can also identify a solid figure by counting the edges .
::您也可以通过计算边缘来识别一个固态数字。

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An edge is the place where two faces meet. Edges are straight; they cannot be curved.
::边缘是两面相交的地方。边缘是直的,不能弯曲。

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Let's look at an example.
::让我们举个例子。

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Count all of the edges on the solid figure above. This figure has 9 edges.
::计算以上固体图案上的所有边缘。 此图有 9 个边缘 。

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Cones, spheres, and cylinders do not have any edges because they do not have any flat sides.
::锥形、球体和圆柱体没有边缘,因为它们没有平面。

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The place where two or more edges meet is called a vertex . A vertex is like a corner. If you have more than one vertex they are called vertices.  You can also count the number of vertices in order to identify .
::两个或两个以上边缘相交的地方称为顶点。顶点就像一个角。如果有一个以上的顶点,则称为顶点。您也可以计算顶点数来识别。

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Let's look at an example.
::让我们举个例子。

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When you count the number of vertices in the figure above, you see there are 7 vertices.  All solid figures that have a vertex at the top are classified as pyramids.
::当您计算上图中的顶点数时, 您可以看到有 7 个顶点。 所有在顶部有顶点的固体数字都被归类为金字塔 。

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This chart gives you an idea of some of the faces, edges of vertices of common solid figures.
::这个图表可以让你了解一些面孔, 以及共同的固体数字的边缘。

Figure Name	Number of Faces	Number of Edges	Number of Vertices
sphere	0	0	0
cone	1	0	0
cylinder	2	0	0
pyramid	at least 4	at least 6	at least 4
prism	at least 5	at least 9	at least 6

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A prism has the same polygonal shape on the top as it has on its base.  A pyramid has a polygonal shape as its base, and a vertex at the top where all of its triangular-shaped side faces meet.
::棱柱顶部的多边形形状与底部的形状相同。 金字塔底部是多边形形状,顶部是顶部的顶部是三角形侧面相交处的顶端。

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Let's look at an example.
::让我们举个例子。

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In the figure above, you can see that the base and the top of the figure are the same polygonal shape. That means that you are working on identifying a type of prism . Let’s use the base to help identify the specific type of prism. The base is a five-sided figure. You know that a five-sided figure is called a pentagon .  This is a pentagonal prism .
::在上图中,您可以看到图的底部和顶部是相同的多边形形状。 这意味着您正在致力于确定一种棱镜类型。 让我们使用基数来帮助确定特定的棱镜类型。 基数是一个五面图。 您知道一个五面图被称为五面图。 这是一个五面棱镜。 这是五面棱镜 。

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Let's look at an example.
::让我们举个例子。

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In the figure above, the base is a polygon and the top forms a vertex.  This means the figure is a pyramid .  The base is a six-sided figure.  You know that a six-sided figure is called a hexagon .  This solid figure is a hexagonal pyramid .
::在上图中,基数是一个多边形,顶部形成一个顶部。这意味着数字是一个金字塔。基数是一个六面图。你知道,六面图被称为六面图。这个固体图是一个六边金字塔。

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When you think about the number of faces, vertices and edges in solid figures, you may start to see patterns emerge.
::当你想一想真实数字中的脸、脊椎和边缘的数量时,你可能会开始看到模式的出现。

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You can see a pattern related to spheres, cones, and cylinders. All of these figures are curved in some way, so they have no edges or vertices. What about their faces? A sphere has no faces, a cone has one circular face, and a cylinder has two circular faces. Therefore, the number of faces increases by one from one figure to the next. This is a pattern.
::您可以看到一个与球体、锥形和圆柱形有关的图案。 所有这些图案都以某种方式曲线, 因此没有边缘或脊椎。 它们的表情如何? 球体没有面孔, 锥形有一个圆形脸, 圆柱形有两个圆形脸。 因此, 面孔的数量会从一个图案到下一个图案增加一个图案。 这是一个图案 。

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What about prisms?  As the number of edges of the base (and parallel top) increases, the number of side faces increases the same amount.  The base of a triangular prism has 3 edges.  It therefore has 3 side faces plus the base and top, or 5 in all. A hexagonal prism has 6 side faces plus the base and top, or 8 faces in all.
::棱晶呢 ? 随着底部( 和平行顶部) 边缘数的增加, 侧面数会增加同样数量。 三角棱晶底有3个边缘。 因此, 底部有3个侧面加上底部和顶部, 或者总共5个。 六角棱晶有6个侧面加上底部和顶部, 或者总共8个面。

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This means that you can put in any number for $\begin{array}{r}n\end{array}$ , representing the number of edges of the base (and top).  You know this also represents the number of faces that make up the sides.  You then add the base as a face and the top as another face, adding two more faces.  Therefore, if the base has  $\begin{array}{r}n\end{array}$  number of edges, then the prism will have  $\begin{array}{r}n+2\end{array}$  number of faces.
::这意味着您可以为 n 输入任意数字, 代表基点边缘( 和顶部) 的数量。 您知道这还代表了构成两边的面孔数量。 然后您将基点添加为一张脸, 顶部添加为另一张脸, 增加两张脸。 因此, 如果基点有 n 边数, 那么棱镜将拥有 n+2 的面孔数量 。

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Now, let's consider pyramids and the pattern that exists here.  As with the prism, the number of edges of the base will equal the number of faces of the triangular sides.  But there is not a top face to add to this since the top is the vertex where the triangular sides meet.  For example, the base of a triangular pyramid has 3 edges.  As with the triangular prism, it therefore has three side faces.  In the case of pyramid, these side faces are triangular with the faces meeting to form the vertex.  You then add the three side faces with the base face to get four faces in total.
::现在,让我们来考虑一下金字塔和这里存在的模式。 和棱镜一样, 底部边缘的数量将等于三角边的面数。 但是没有顶部面面可以加到这个上方, 因为顶部是三角边相交的顶部。 例如, 三角金字塔的底部有 3 个边缘。 因此, 三角棱柱有 3 个侧面。 在金字塔的情况中, 这些侧面面与面部会合的三角面面面会形成顶部。 然后您加上三面面面加上底面面, 总共得到 4个面孔 。

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This means that you can put in any number  $\begin{array}{r}n\end{array}$ , representing the number of edges of the base.  You know this also represents the number of triangular faces that make up the sides.  You then add the base as a face, adding one more face. Therefore, if the base has  $\begin{array}{r}n\end{array}$  number of edges, then the pyramid will have  $\begin{array}{r}n+1\end{array}$  number of faces.
::这意味着您可以输入任何 n 数字, 代表基点边缘的数量。 您知道它也代表构成边边的三角面数。 然后您将基点添加为面, 并添加一个面。 因此, 如果基点有 n 边数, 那么金字塔将拥有 n+ 1 的面数 。

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Examples
::实例

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Example 1
::例1

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Earlier, you were given a problem about Martha and the Louvre.
::之前,你得到一个问题 关于玛莎和卢浮宫。

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She notes that the base of the structure has 4 edges.  Using this information, what is the name of this figure and how many faces does it have?
::她注意到该结构的底部有四个边缘。 使用此信息, 此图的名称及其有多少面孔 ?

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First, you know that the number of edges of the base is the same number as that of the side faces.  There are 4 side faces.
::首先,你知道基底边缘的数量与侧面的相同。有4张侧面。

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Next, see that there is a vertex at the top and notice that the side faces are triangles.  You know this is a pyramid.
::接下来,请看顶部有一个顶端的顶端, 并注意侧面是三角形。 你知道这是一个金字塔。

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Then, use the formula to calculate the number of faces of a pyramid ( $\begin{array}{r}n+1\end{array}$ ):  4 + 1 = 5
::然后,使用公式计算金字塔(n+1)面数:4+1=5

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The answer is that the Louvre Pyramid is a square pyramid with 5 faces.
::答案是卢浮宫金字塔是五张脸的平方金字塔

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Example 2
::例2

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Name the figure and the number of faces.
::列出图形和面孔数。

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First, observe that the side faces of the figure are all triangular in shape forming a vertex at the top.  You know this is a pyramid.
::首先,观察图的侧面是三角形的, 形成顶部的顶部顶部的顶部。 你知道这是一个金字塔。

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Next, see that the base is a hexagon with 6 edges.
::接下来,看基座是六边六边的六边形。

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Then, using the formula   $\begin{array}{r}n+1\end{array}$ to calculate the number of faces of a pyramid , calculate the number of faces in this figure as 7 in total.
::然后,使用公式 n+1 来计算金字塔的面数, 计算数字中的面数为 7 。

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The answer is a hexagonal pyramid with 7 faces.
::答案是六边形金字塔,有7张面孔。

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Here is another one:
::这是另一个:

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A figure has a base and parallel top face, each with 7 edges. How many faces does it have?
::数字有一个基面和平行的顶面面,每个面部有7个边缘。它有多少面部?

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First, if the base has 7 edges, there must be 7 side faces.  
::首先,如果基地有7个边缘, 必须有7个侧面的面孔。

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Next, since there is a parallel top face, you know this is a prism.
::接下来,因为有一个平行的顶面脸, 你知道这是一个棱镜。

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Then, you use the formula for calculating the number of faces in a prism:
::然后,你用公式来计算棱镜面数:

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::n+2 = 面数7+2=9

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The answer is that the prism has 9 faces.
::答案是,棱镜有9张面孔。

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Name the figure and the number of faces.
::列出图形和面孔数。

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Example 3
::例3

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First, see that the base and the top are parallel, each with 9 edges.  Since there is a base with a parallel top, you know this is a prism.  You also know that a 9-sided figure is a nonagon .
::首先, 请看底部和顶部是平行的, 每个边都有 9 个。 因为底部有一个平行的底部, 您知道这是一个棱镜 。 您也知道 9 面的图是非 字形 。

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Next, you know that if the base has 9 edges, there are 9 side faces.
::接下来,你知道,如果基地有9个边缘, 有9个侧面的面孔。

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Then, you use the formula for calculating the number of faces of a prism:  
::然后,你使用公式计算棱晶面数:

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The answer is this solid figure is a nonagonal prism with 11 faces. 
::n+2 = 面孔数 9+2 = 11 。 答案是, 此固态数字是一个有 11 面孔的非教义棱镜 。

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Example 4
::例4

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First, note that the side faces of the figure are all triangular, creating a vertex at the top.  You know this is a pyramid.
::首先,请注意图的侧面都是三角形, 在顶部创建了顶部的顶点。 你知道这是一个金字塔。

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Next, observe that the base is a septagon with 7 edges.
::接下来,观察基地是7边缘的。

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Then, use the formula to calculate the number of faces in the figure.
::然后,使用公式计算图中面孔的数量。

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::n+1 = 金字塔7+1=8面孔中的面孔数

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The answer is the figure is a septagonal pyramid with 8 faces.
::答案是这个数字是一个有8张面孔的西伯利亚金字塔。

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Example 5
::例5

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A prism with a base of a pentagon
::具有五角形基底的棱晶

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First, you know that the base is a pentagon with 5 edges.
::首先,你知道基地是五边的五角形

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Next, you know that the prism has a base with a parallel top.
::接下来,你知道棱镜有一个基底 与平行的顶部。

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Then, calculate the number of faces using the formula for a prism ( $\begin{array}{r}n+2\end{array}$ ):
::然后,使用棱晶(n+2)的公式计算面孔数 :

5 + 2 = 7

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The answer is this solid figure is a pentagonal prism with 7 faces.
::答案是这个坚固的数字 是一个有7张面孔的五角形棱镜

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Review
::回顾

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Count the number of faces, edges, and vertices in each figure.
::计算每个图中的脸、边缘和顶点的数量。

1.  
   
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 

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Answer each question.
::回答每个问题。

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13. A figure has flat faces. What kind of figure is it?
    ::数字是平面的,是什么样的数字?
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14. A figure has one pair of parallel sides that are circular. What kind of figure is it?
    ::数字有一个平行的两面,是圆形的。这是什么样的数字?
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15. Decagons are polygons that have ten sides. How many faces does a decagonal prism have?
    ::Decagons是多边形,有十面。一个计价棱镜有多少面孔?
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16. A hexagon has six sides. How many faces does a hexagonal prism have?
    ::六边形有六面形。六边形棱镜有多少面孔?
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17. A heptagon has seven sides. How many faces does a heptagonal prism have?
    ::七面形有七面形,七面形棱镜有多少面孔?

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Review (Answers)
::回顾(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 。

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Resources
::资源