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  • 固体交叉区

    Cross Sections
    ::交叉科

    A cross section is the intersection of a figure in three-dimensional space with a plane . It is the face you obtain by making a "slice" through a solid object. A cross section is two-dimensional. The figure (face) obtained from a cross section depends upon the orientation (angle) of the plane doing the cutting. Consider the following hexagonal pyramid .
    ::十字部分是三维空格图与平面的交叉点。它是您通过固态物体制作“切片”获得的面孔。交叉部分是二维的。从十字部分获得的图(面)取决于切割的平面的取向(角)。请考虑以下六边形金字塔。

    A hexagonal pyramid showing a plane slice and its cross section as a hexagon.

    Cross sections perpendicular to the base and through the vertex will be triangles. Below, you can see a plane cutting through the pyramid, part of the pyramid removed, and the cross section.
    ::垂直于基底的横截段和穿过顶端的横截段将是三角形。下面,您可以看到一架飞机穿过金字塔,金字塔的一部分被清除,横截段被切开。

    Vertical Cross Sections
    Vertical Cross Sections

    You could also take a slice parallel to the base. Cross sections parallel to the base will be hexagons.
    ::您也可以在基座上平行切片。 与基座平行的截段将是六边形 。

    It is also possible to take cross sections using planes that are neither parallel nor perpendicular to the base. These can be much more difficult to visualize. Physical models or dynamic geometry software are extremely helpful. For example, the same hexagonal pyramid has been sliced at a slant below. The cross section is a hexagon .
    ::也可以使用与基地既不平行也不垂直的平面截取横截面。 这可能更难想象。 物理模型或动态几何软件非常有用。 例如, 相同的六边形金字塔被切入下面的倾斜线。 横截面是六边形 。

    Cross sections are one way of representing three-dimensional objects in two dimensions .
    ::交叉区域是代表两个维度的三维天体的一种方式。

    CK-12 Interactive:  Cross Sections and Solids
    ::CK-12 互动:交叉区块和固体

    Cross Sections of Solids
    ::固体交叉区

    Explore the different cross sections of the solids given below.
    ::探讨下述固体的不同交叉部分。

    Solid: Cylinder Cross Section: Horizontal
    ::固体:圆柱交叉部分:横向

    Constructing Cross Sections
    ::构造交叉区域

    Using planes parallel to the base, what cross sections can you construct from a rectangular prism ?
    ::使用与基底平行的平面, 您可以用矩形棱柱构造哪些交叉区域 ?

    Considering only planes that are parallel to the base, all cross sections are . This is shown below.
    ::仅考虑与基地平行的飞机,所有横截段均为.如下所示。

    Using planes perpendicular to the base, what cross sections can you construct from a rectangular prism?
    ::使用与基底垂直的平面, 您可以用矩形棱柱构造哪些交叉区域 ?

    Considering only planes that are perpendicular to the base, all cross sections are rectangles. This is shown below.
    ::仅考虑与基地垂直的飞机,所有交叉区段都是矩形,如下所示。

    Describing Cross Sections
    ::规定跨科

    1. A rectangular pyramid is sliced by a plane parallel to its base. Describe the cross section. How does moving the plane so that it is closer or further from the base change the cross section?
    ::1. 长方形金字塔用与底部平行的平面切除,描述横截面。如何移动平面使其靠近或离底部更远?

    The cross section will always be a rectangle . The closer the plane is to the base, the bigger the rectangle.
    ::横截面将永远是一个矩形。 飞机越接近底部, 矩形越大 。

    2. Describe how to create a triangular cross section of a rectangular pyramid.
    ::2. 描述如何创建矩形金字塔三角横截面。

    Slice the pyramid with a plane that is perpendicular to the base of the pyramid.
    ::用直通金字塔底部的平面切开金字塔

    Cross Sections of Solids - Examples
    ::固体的交叉区块 -- -- 实例

    Example 1
    ::例1

    A pentagonal prism is sliced with a plane parallel to the base. The area of the cross section is 15 square inches. If the height of the prism is 20 inches, what is the of the prism?
    ::五角棱柱被切除为与基底平行的平面。横截段的面积是15平方英寸。如果棱柱的高度是20平方英寸,棱柱的高度是多少?

    A pentagonal prism is sliced with a plane parallel to the base. This means that the cross section is a pentagon that is congruent to the base. The area of the cross section is 15 square inches. This means that the area of the base is 15 square inches. Since the height of the prism is 20 inches, the volume of the prism is:
    ::五角棱柱被切除为与基底平行的平面。 这意味着横截面是一个与基底相同的五角形。 横截面的面积是 15 平方英寸。 这意味着基底的面积是 15 平方英寸。 由于棱柱的高度是 20 英寸, 棱柱的体积是 :

    V = A base ⋅ h V = ( 15 ) ( 20 ) V = 300   in 3
    ::V=AbasehV=(15)(20)V=300英寸3

    Example 2
    ::例2

    Describe the cross sections of a cylinder that are perpendicular or parallel to the base of the cylinder.
    ::描述圆柱体垂直或与圆柱体底部平行的交叉部分。

    R ectangles and circles, respectively. 
    ::分别是矩形和圆圈。

    Example 3
    ::例3

    Could the cross section of a cylinder be an ellipse (oval)?
    ::圆柱体的交叉部分会不会是椭圆形(卵巢)?

    Yes. If the slice is slanted as shown below:
    ::是。如果切片倾斜如下:

    A slanting cross section of a cylinder showing an elliptical shape.

    Example 4
    ::例4

    A cylinder is sliced parallel to its base. The area of the cross section is 36 π   in 2 . What is the radius of the cylinder?
    ::圆柱体与基座平行切片。 横角区域是 36 英寸 2 。 圆柱体的半径是多少 ?

    The cross section of the cylinder is a circle .
    ::圆柱体的横截面是一个圆形。

    The area of cross section is:
    ::跨节领域是:

    A = π r 2 36 π = π r 2 36 = r 2 6 = r
    ::r236r236=r26=r

    Therefore, the radius of the circle = 6 inches.
    ::因此,圆的半径=6英寸。

      Summary
    • Cross sections are the intersection of a figure in three-dimensional space with a plane, resulting in a two-dimensional face.
      ::交叉部分是三维空间中数字与平面的交叉点,产生二维面孔。
    • The figure obtained from a cross section depends on the orientation of the plane doing the cutting.
      ::从横截段获得的数字取决于切割飞机的方向。
    • The area of a cross section can be used to determine the volume of a prism, as the volume is equal to the area of the base multiplied by the height.
      ::横截段的面积可用于确定棱晶的体积,因为体积等于基数乘以高度的面积。

    Cross Sections of Solids - Review Questions
    ::固体交叉部分 -- -- 审查问题

    1. Describe the cross sections of a sphere.
    ::1. 描述一个领域的各跨部分。

    2. Describe the cross sections of a pentagonal prism that are parallel or perpendicular to the base.
    ::2. 描述与底部平行或垂直的五角形棱镜的交叉部分。

    3. Describe the cross sections of a pentagonal pyramid that are parallel or perpendicular to the base.
    ::3. 描述五角形金字塔中与基地平行或垂直的交叉部分。

    4. Could the cross section of a cube be a triangle? Explain.
    ::4. 立方体的横截面能否是三角形?

    5. A cross section of a pyramid is taken parallel to its base. What are the connections between the cross section and the base?
    ::5. 金字塔的横截面与其底部平行,横截面与底部之间有何联系?

    6. A cross section of a pyramid is taken perpendicular to its base. What shape is the cross section? Does the shape of the base matter?
    ::6. 金字塔的交叉部分与它的底部相连,交叉部分的形状是什么?基面的形状是否重要?

    7. For a prism, information about what type of cross section can help you to determine the volume?
    ::7. 就棱镜而言,关于哪类横截面的信息可以帮助您确定数量?

    8. A cylinder with radius 4 inches is sliced parallel to its base. What is the area of the cross section?
    ::8. 半径为4英寸的圆柱体被切除与其底部平行。

    9. The volume of a cylinder is 360 π   i n 2 . The cylinder is sliced parallel to its base. If the height of the cylinder is 10 inches, what is the area of the cross section?
    ::9. 圆柱体的体积是360英寸英寸,圆柱体的体积是360英寸,圆柱体与基数平行切片,如果圆柱体的高度是10英寸,圆柱体的面积是多少?

    Use the solid below for 10-12.
    ::10 -12用下方的固体

    3D representation of a pentagonal pyramid above a rectangular prism with labeled dimensions.

    10. A plane slices the solid parallel to its base. Describe the cross section. How does the cross section change as the plane moves further from the base?
    ::10. 平面切除与基底平行的固体,描述横截段。 当平面从基底进一步移动时,横截段如何变化?

    11. A plane slices the solid perpendicular to its base through the center of the base. Describe the cross section.
    ::11. 一平面通过基底中心切断与基底垂直的固体,描述横截面。

    12. Find the area of the cross section from #11.
    ::12. 从 #11找到横截段的面积。

    Use the solid below for 13-15.
    ::13 -15使用下面的固体

    A cone and cylinder with dimensions: 20 in height, 6 in radius, and 13 in cylinder height.

    13. A plane slices the solid parallel to its base. Describe the cross section. How does the cross section change as the plane moves further from the base?
    ::13. 平面切除与基底平行的固体,描述横截段。 当平面从基底进一步移动时,横截段如何变化?

    14. A plane slices the solid perpendicular to its base through the center of the base. Describe the cross section.
    ::14. 一平面通过基底中心切断与基底垂直的固体,描述横截面。

    15. Find the area of the cross section from #14.
    ::15. 从#14找到横截段的面积。

    16. Find the area of the cross-section below. (The isometric grid paper is composed of equilateral triangles whose sides are of unit length 1) How does the area of this cross section compare with the parallel faces? Are there other cross-sections with areas the same as the parallel faces? Describe them and find their area.
    ::16. 找出以下横截面的面积。 (等度网格纸由两边为单位长度的等边三角组成,1) 这个横截面的面积与平行面相比如何比较?是否有其他交叉面的面积与平行面的相同?描述和找到其面积。

    Cross-section of a rectangular prism shown on an isometric grid, labeled with point F.

    17. Use isometric grid paper, interactive geometry software , or a free-hand sketch to investigate the following questions, and support your answers with drawings and explanations. Given a rectangular prism:
    ::17. 使用实测网格纸、交互式几何软件或自由手动草图调查下列问题,用图纸和解释支持回答。

    1. How many cross sections are parallel to bases?
      ::有多少跨区与基地平行?
    2. How many cross sections intersect two complete edges of the prism?
      ::有多少横截面横交两个完整的棱镜边缘?
    3. How many cross sections intersect one complete edge of the prism?
      ::有多少横截面交叉于棱镜的完整边缘之间?
    4. Is it possible to create a cross-section that intersects only one vertex of the prism?
      ::是否有可能建立一个交叉体,只交叉一个棱镜的脊椎?
    5. Is it possible to create a cross-section that intersects no vertices of the prism?
      ::能否建立一个横截面,将棱晶的脊椎交叉开来?
    6. Is it possible to create a cross-section that intersects two vertices but no complete edges?
      ::有没有可能建立一个交叉体, 将两个脊椎交叉, 但没有完整的边缘?
    7. Is it possible to create a cross-section that passes through only three faces? Four faces? Five faces?
      ::能否建立十字路口,只穿过三个面孔,四个面孔,五个面孔?
    8. What is the maximum number of faces a cross-section can pass through ?
      ::交叉剖面能够穿透的最大面孔数是多少?
    9. What is the relationship between the number of faces a cross-section passes through, and the number of sides of the resulting polygon?
      ::横截面通过面数与由此形成的多边形的侧面数之间的关系是什么?

    18. The following pyramid has a rectangular base. Find the area of the shaded cross-section below. Round your answer to the nearest hundredths place.
    ::18. 下列金字塔有一个矩形基。 请在下面找到阴影横截面的面积。 将您的答案转至最近的一百个位置 。

    A triangular pyramid with labeled points and edges, including shaded area and dimensions.

    19. Sketch a cone and answer the following questions. Explain your answers.
    ::19. 绘制圆锥并回答下列问题:解释你的答复。

    1. Sketch the shape of a cross-section that is not parallel to the base and only intersects the lateral face of the cone.
      ::绘制与底部不平行的横截面的形状,只交叉锥锥体的横向面。
    2. What is the shape of a cross-section that is parallel to the base?
      ::与基底平行的横截面的形状是什么?
    3. What is the shape of a cross-section that is perpendicular to the base and intersecting the apex of the cone?
      ::与锥体底部相连的交叉剖面的形状是什么? 锥体顶部交叉的横截面的形状是什么?
    4. Sketch and describe the shape of a cross section that is perpendicular to the base but doesn't intersect the apex.
      ::并描述与底部垂直但不会交叉的横截面的横截面的形状。

    Review (Answers)
    ::审查(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 。