Section outline

  • In This Lesson
    ::在本课程中

    Y ou will be finding the surface areas of different types of pyramids .  Depending on the base, the triangular faces may or may not be congruent .  If the base is a square , for example, then all 4 of the triangular faces will be congruent.  However, if the base is a scalene right triangle , then the 3 triangular lateral faces will be different.
    ::您将会找到不同类型金字塔的表面区域。 取决于底座, 三角形面貌可能或可能不相容。 例如, 如果基座是一个正方形, 那么三角形面貌的所有4个面孔都是正方形的。 但是, 如果基座是一个右三角形, 那么三角形面貌将会不同 。

    Nets of different pyramids showing their geometric shapes and corresponding layouts.
    Pyramid Nets

    The Louvre Pyramid
    ::卢浮金字塔

    The Louvre Museum in Paris, France has a unique entrance.  The entrance is a 71 foot tall square based pyramid made entirely of glass.  The square's edges are all 112 feet and the slant  height is 90.4 feet.  How much glass was needed to construct this pyramid?
    ::法国巴黎卢浮宫博物馆有一个独特的入口。 入口是一个71英尺高的方形金字塔, 完全由玻璃制成。 广场的边缘是112英尺,斜坡高度是90.4英尺。 建造这个金字塔需要多少玻璃?

    The Louvre Pyramid in Paris, illuminated at dusk, showcasing its glass structure.
    The Louvre Museum

      In order to determine how much glass is needed, you need to find the lateral (do not include the area of the square base). The interactive for this problem will help you determine the difference between the true height and the slant height of the pyramid. Use the slider to see the triangle created by these two heights and the base. Once you are comfortable with how the two heights are related, you can unfold the pyramid into its and find the lateral surface area.
    ::为了确定需要多少玻璃, 您需要找到平面( 不包括平方的面积) 。 这个问题的交互作用将帮助您确定金字塔真实高度和倾斜高度之间的差别。 使用滑动器查看这两个高度和基数所创建的三角形。 一旦您对这两个高度的关系感到舒服, 您就可以将金字塔展出到它的平面区域, 并找到平面区域 。

     

     

    INTERACTIVE
    Unfolding the Louvre
    minimize icon
    • Drag the sliders to see the difference between true height and lateral height as well as unfold the net of a square pyramid.

      ::拖动滑动符以看到真实高度和横向高度之间的差别,并展开平方金字塔的网。
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    Each of the lateral faces of the Louvre pyramid is made up window panes in the shapes of rhombi (a quadrilateral with 4 congruent sides, such as a square or a kite) and .  How should the rhombi relate to the equilateral triangles so that there is no gap in between panes?  Why do you suppose the designer chose to split up the faces in this way?  
    ::卢浮宫金字塔的每个横向面部都是以 rhombi (一个四边形,四面形,如方形或风筝) 和 。 卢浮宫金字塔的每个横向面部都是由窗帘组成的。 卢浮宫的每个面部应该如何与等边三角形联系起来, 从而不使窗帘间出现差距? 为什么你认为设计师会选择这样分割面部呢 ?

    Discussion Question
    ::讨论问题

    If the base of a pyramid has four congruent sides, the four triangles that make up the sides of the pyramid will always be congruent. Mercedes claims that if the base of the pyramid is a rhombus this will not be true. Is Mercedes correct? Use evidence to support your argument . What tools could you use to answer this question?
    ::如果金字塔的底部有四个相容的两面,构成金字塔两侧的四个三角将永远是相同的。 奔驰声称, 如果金字塔的底部是圆柱形, 则事实并非如此。 奔驰是正确的吗? 使用证据支持你的论点。 您可以用什么工具回答这个问题 ?

    Making a Fort
    ::建立堡垒

    Hunter and her brother Cooper want to make a fort out of material  and PVC pipe.  They decide on a pyramid shape.  In order for it to fit in a particular space in the corner of Hunter's room, the base must look like this:
    ::Hunter和她的兄弟Cooper想用物质和聚氯乙烯管道做一个堡垒。 他们决定一个金字塔形状。 为了让它适应杭特房间角落的一个特定空间, 基地必须像这样:

    A triangular shape with labeled dimensions for constructing a fort.
    Base of the fort

    In addition, the height is going to be directly over the right angle in the corner.  Cooper wants the true height to be at least 5 feet so he can stand up in the fort.  The height of the other two faces will be shown in the interactive and depends on the true height.   Hunter has 4 blankets totaling 5760 sq. in.  What is the maximum height the fort can be, given the following restrictions?
    ::此外,高度将直接在角角的右角上方。 Cooper希望真正的高度至少为5英尺,以便他能够站到堡垒上。 另外两张脸的高度将在互动中显示,并取决于真正的高度。 Hunter有四张毯子,总共在5,760平方米处。 根据以下的限制,堡垒能达到的最高高度是多少?

    • The ceiling height is 8 feet.
      ::天花板的高度是8英尺
    • The minimum height of the pyramid must be 5 feet.
      ::金字塔的最低高度必须是5英尺
    • The maximum surface area is 5760 sq. in.
      ::最大表面面积为5,760平方米。
    • The height of the two faces with base of 24 inches is the true height of the pyramid.
      ::金字塔的顶峰是金字塔的顶峰 以24英寸为基点的两张脸的顶峰是金字塔的顶峰
    • You do not need to find the length of the PVC pipe needed. 
      ::您不需要找到PVC管道所需的长度 。
    • Interactive will show you both slant heights and the total surface area as the true height changes.
      ::随着真实高度的变化,互动将显示倾斜高度和总表面面积。
    • If you need to unfold the pyramid to see the net, click the unfold button in the interactive.
      ::如果您需要展出金字塔查看网,请在互动中单击展出按钮。

     

    INTERACTIVE
    Blanket Fort
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    • Drag the red point to change the height of the pyramid.

      ::拖曳红色点以改变金字塔的高度 。
    • Click the "Unfold" button to see the net of this pyramid.
      ::单击“ 翻转” 按钮查看此金字塔的网 。
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    T he formula for surface area of a pyramid is simply to add up the area of all of the sides. Recall that lateral surface area does not include the area of the base.
    ::金字塔表面面积的公式只是将两边的面积相加。 回顾平面面积不包括基底面积。

       Summary
    ::摘要

    • To find the surface area of a pyramid, add up the areas of all the faces.
      ::要找到金字塔的表面积, 将所有面孔的面积加在一起。
    • If the base is a square or rhombus, all 4 of the lateral, triangular faces will be congruent.
      ::如果基座是正方形或正方形,则所有4个平面、三角面面面将一致。
    • If the base is a parallelogram or rectangle, there will be 2 pairs of congruent lateral faces.
      ::如果基座是平行图或矩形,则有两对相近的横向面孔。
    • If the base is an equilateral triangle, then all 3 of the lateral, triangular faces will be congruent.
      ::如果基底是一个等边三角形, 那么所有3个平面, 三角面将是一致的 。