Section outline

  • In This Lesson
    ::在本课程中

    You will find dimensions for different types of cross sections that can be generated from prisms, cylinders, pyramids , and cones. The specifics depend on if the cross section is parallel or perpendicular to the base (this lesson will not include dimensions of cross sections on an angle).
    ::您会发现来自棱晶、圆柱、金字塔和锥锥体的不同类型跨区段的尺寸。 具体细节取决于跨区段是否与底部平行或垂直( 本课不会包括角度上的跨区段的尺寸 ) 。

    Can You Cut It?
    ::你能剪掉吗?

    Use the interactive to slice a  rectangular prism , a quadrilateral prism (no sides of the bases are congruent or parallel), and a triangular prism using planes that are perpendicular and parallel to the base. Try to determine the dimensions of the cross sections.
    ::使用互动来切除矩形棱柱、四边棱柱( 基点的两边没有相似或平行) , 以及三角棱柱, 使用垂直和与基点平行的平面。 尝试确定横截段的尺寸 。

    INTERACTIVE
    Can You Cut It?
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    • Use the dropdown menus to change the prism and the type of cross section plane.
      ::使用下拉菜单来改变棱镜和跨区平面的类型。
    • Move the red slider to move the cross section plane.
      ::移动红色滑块以移动横截面平面 。
    • Move the blue slider to change the opacity of the prism.
      ::移动蓝色滑块以改变棱晶的不透明性。
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    1.

    Consider the cross sections of the quadrilateral and triangular prism. Which statements are true if the cross section is perpendicular to the base?
    ::考虑四边形和三角棱镜的跨部分。 如果跨部分与底部垂直, 哪种语句是真实的 ?

    Select all that apply
    a

    The cross section will be the same shape as the base.
    ::横截段的形状将与基数相同。

    b

    The cross section will be a rectangle.
    ::横截段将是一个矩形。

    c

    One dimension of the cross section will be the height of the prism.
    ::横截段的一个方面将是棱镜的高度。

    d

    The cross section will be the same size as the base.
    ::横截段的大小将与基数相同。

    A prism is a polyhedron made up of two congruent, parallel faces (the bases) and sides that are (lateral faces). For all prisms, cross sections that are parallel to the base are congruent to the base. Rectangular prisms have cross sections that are congruent to any face they are parallel to. Congruent means that the dimensions will be exactly the same.
    ::棱镜是一个多面体,由两个相近、平行面(基)和侧面(边面)组成。 对于所有棱镜,与底面平行的交叉区块与底部一致。矩形棱柱的交叉区块与任何面块相似。共容意味着维度将完全相同。

    Cross sections that are perpendicular to the base only have the same height as the prism. Be careful when determining which dimension is the height of a prism. The height of a prism is the height of the lateral faces . Make sure you never use a dimension from the base.
    ::垂直于底部的交叉部分只有与棱柱相同的高度。 当确定一个棱柱的高度时要小心。 棱柱的高度是横向面部的高度。 请确定您从不使用底部的尺寸 。

    Can You Cut It? Part 2
    ::你能剪剪吗?

    Use the interactive to slice a rectangular pyramid , a quadrilateral pyramid (no sides of the base are congruent or parallel), and a triangular  pyramid. Try to determine the dimensions of the cross sections. If the cross sections are not congruent to a face, can you determine its dimensions?
    ::使用互动来切除矩形金字塔、 四边金字塔( 基底的两边没有相似或平行) 和三角金字塔 。 尝试确定交叉区段的维度。 如果交叉区段与脸部不相容, 您能否确定它的维度 ?

    INTERACTIVE
    Can You Cut It? Pt.2
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    • Use the dropdown menus to change the pyramid and the type of cross section plane.
      ::使用下拉菜单来改变金字塔 和跨段平面的类型
    • Move the red slider to move the cross section plane.
      ::移动红色滑块以移动横截面平面 。
    • Move the blue slider to change the opacity of the pyramid.
      ::移动蓝色滑块以改变金字塔的不透明性 。
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    As you can see, when the cross sections are parallel to the base, they will have the same shape as the base, but will be smaller.  They are proportional to the base.  If the cross section is perpendicular to the base, it could have the same height if it passes through the vertex .  If the pyramid has a rectangular base, then the base of the cross section will be the same as the edge of the base that it is parallel to.
    ::可以看到, 当交叉区域与基底平行时, 它们与基底的形状相同, 但其大小则较小。 它们与基底成正比。 如果交叉区域与基底垂直, 如果穿过顶部, 它可能具有相同的高度。 如果金字塔有一个矩形基, 则交叉区域的底部与基底的边缘相同。

    In the final interactive, you will work backwards to determine the type of polyhedron a given cross section could come from.
    ::在最后的交互式互动中,您将倒着工作,以确定某一截面段可能来自哪类聚希德龙。

    From 2D to 3D
    ::2D至3D之间

    Given a 2-dimensional cross section, could you determine the polyhedron it comes from? In this interactive, you will be given cross sections that are parallel or perpendicular to the the base of a prism or pyramid. You will need to determine the type of polyhedron that the cross section comes from. Note that there could be more than one correct answer.
    ::给定一个 2 维的交叉区域, 您能否确定它来自的多面体 ? 在此互动中, 您将得到与棱柱或金字塔基底平行或垂直的交叉区域 。 您需要确定该交叉区域来自哪类多面体 。 请注意, 可能有一个以上的正确答案 。

    INTERACTIVE
    Cross Section
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    • Check the box for each solid that has a cross section of the given shape.
      ::检查每个有给定形状交叉部分的固体的框。
    • Use the "Check" button to check your answer. 
      ::使用“ 检查” 按钮检查您的回答 。
    • Use the drop down menu to switch between different shapes for cross sections.
      ::使用下拉菜单在横截段的不同形状之间切换。
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    Discussion Question
    ::讨论问题

    What strategies did you use to connect cross sections to ? Were there any patterns that you noticed?
    ::你用什么策略连接交叉区段?

    Extension: Cross Sections of Spheres
    ::扩展范围:球体交叉部分

    Determine all the cross sections of a . Are there any besides a circle ? Consider when a  plane just touches one point of the sphere .
    ::确定 a 中的所有跨区 。 除了圆外, 是否有其他区域 ? 考虑一平面何时触及球体的某个点 。

       Summary
    ::摘要

    • Cross sections that are parallel to the face of a rectangular prism are also congruent to that face.
      ::与矩形棱镜面平行的交叉部分也与该面相类似。
    • Cross sections that are parallel to the base of a prism are also  congruent to that base.
      ::与棱镜基础平行的交叉部分也与该基础一致。
    • Cross sections that are parallel to the base of a pyramid  are proportional to the base. 
      ::与金字塔底部平行的交叉部分与基数成比例。
    • The maximum height of a cross section  that is perpendicular to the base of  a pyramid is the height of the pyramid.
      ::与金字塔底部垂直的横截段的最大高度是金字塔的高度。