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  • 立方体和矩形棱晶体表面面积

    In  This Lesson
    ::在本课程中

    You will learn how to find the of a cube and rectangular prism . Now, you will use what you know about finding the area of a two-dimensional shape and applying it to a three-dimensional one.
    ::您将学会如何找到立方体和矩形棱柱。 现在, 您将使用您所知道的 找到二维形状的区域, 并将其应用到三维形状 。

    Discussion Questions
    ::讨论问题 讨论问题

    1. James wants to paint his house. The gallon of paint he bought says that it will cover 400 square feet. If James' house is 3-dimensional and the paint can is 3-dimensional, why is the paint coverage measured in the two-dimensional square feet?
      ::James想画他的房子。他买的一加仑的油漆表示它将覆盖400平方英尺。如果James的房子是三维的,油漆罐是三维的,为什么用两维平方英尺测量的油漆覆盖率?
    2. How could James determine if he bought enough paint?
      ::詹姆士怎么能确定 他买的油漆够不够?

    Chocolate Poofs Cereal
    ::谷类巧克力

    The surface area of a polyhedron is the sum of the areas of all of its faces, including any bases. The lateral surface area is only the surface area of the lateral faces and does not include the surface area of the base(s).
    ::多希德龙的表层面积是其面部(包括任何基地)的总和,平面面积只是横向面面的表层面积,不包括基面的表层面积。

    Corey and Kate are developing a new cereal, Chocolate Poofs. They need to design the box so that it is eye-catching for kids, but also informative for parents. A typical cereal box is a rectangular prism, with dimensions 8 in x 2 in x 12 in. How would you figure out how much surface area they have to put all the vital information?
    ::科里和凯特正在开发一个新的麦片,巧克力球。 他们需要设计这个盒子, 以便让孩子能够捕捉眼睛, 但也让父母了解情况。 一个典型的麦片盒是一个长方形棱柱, 尺寸为8x2x12英寸。 您将如何知道他们需要多少表面积才能提供所有重要信息 ?

    • The cereal box in the interactive already has the correct dimensions. You need to determine the surface area.
      ::互动中的麦片盒已经具有正确的尺寸。 您需要确定表面积 。
    • Slide the Open/Close slider so that it "opens" the box and you can see all the surfaces.
      ::滑动 Open/Close 滑块, 以便“ 打开” 框, 您可以看到所有表面 。
    • Determine the area of each rectangle and add them together to get the total surface area.
      ::确定每个矩形的面积,并将其加在一起,以获得总表面面积。

     

    Covering a Rubik's Cube
    ::覆盖魔方

    The Rubik's Cube is a cubic puzzle made up of 26 "cubelets."  The goal of the puzzle is to arrange the cubelets so that each of the 6 faces of the cube  contains the same color.  When the cube is solved, what is the surface area of each color?  What is the surface area of the whole cube?  Each of the six faces will be one color but split into 9 smaller squares of: white, yellow, orange, green, blue and red. The length of each edge is 2.25 inches.
    ::魔方是一个由 26 个“ 立方体” 组成的立方谜。 谜方的目标是排列立方体的立方体, 使立方体的6 张面都包含相同的颜色。 当立方体解开时, 每个颜色的表面面积是多少? 立方体的表面面积是多少? 六张面的每个颜色都是一个颜色, 但分成9个小方形: 白色、 黄色、 橙色、 绿色、 蓝色 和 红色。 每个边缘的长度是 2. 25 英寸 。

    Use the interactive to simulate the Rubik's cube and try to solve it. 
    ::使用交互式程序模拟魔方的立方体, 并尝试解答它 。

    Tipsy Tower!
    ::堤普斯塔!

    The game Tipsy Tower  is a game made up of 54 blocks and starts with them set up as a rectangular prism, measuring 27 cm tall, with a square base. Each block has a set length, where the width is one-third of the length, and the height is one-fifth of the length. W hat are the dimensions of the block?  What are  the dimensions of the starting prism ?
    ::Tipsy Tower 游戏由54个区块组成, 以它们作为长方形棱柱开始, 长27厘米高, 以平方基。 每个区块都有固定长度, 宽度是长度的三分之一, 高度是长度的五分之一。 区块的尺寸是多少? 起始棱柱的尺寸是多少 ?

    • Configure the block to have dimensions that reflect the description above.  Move the sliders for width, height, and length to change the shape of the block.
      ::配置区块以配置反映以上描述的尺寸。 移动宽度、 高度和长度的滑动符以改变区块的形状 。
    • Once you have the correct shape of the block, stack them so that they measure 27 cm tall and there are 54 total.  How many levels of blocks are there?
      ::一旦你拥有了正确的区块形状, 把它们堆叠起来, 这样它们就能测量到27厘米高, 总共是54个区块。 有多少个区块?

    Hints:
    ::提示 :

    1. The  number of blocks that make up the height is a factor of 54.
      ::构成高度的区块数为54。
    2. All three dimensions of each block are multiples of 0.5 cm.  
      ::每个区块的所有三个维度都是0.5厘米的倍数。

       Summary
    ::摘要

    • In general, the surface area of a polyhedron is the sum of the areas of all of the faces. 
      ::一般来说,一面形的表面积是所有面孔面积的总和。
    • For a rectangular prism, with height h ,  width w ,  and length l ,  the formula for surface area is  2 ( l ⋅ w ) + 2 ( l ⋅ h ) + 2 ( h ⋅ w ) .  
      ::对于具有高度h、宽w和长度l的矩形棱柱,表面积的公式为2(lw)+2(lh)+2(hw)。
    • For a cube with sides of length s, the surface area is: 6s 2 .
      ::对于具有长度两边的立方体,表面面积为: 6s2。