GRE Geometry | Three – Dimensional Figures

An object which has only height and length are called 2-dimensional figures while an abject which has height length and width are called 3-dimensional object.

Examples of 2-D figures:

Examples of 3-dimensional figures:

  1. Cube:

    In a cube length breadth and height are all equal.

    length = breadth = height 

    let’s length, breadth and height as a.
    It has 6 square face each with side length a.



    Volume of cube = a * a * a
    Volume of cube = a3
    
    Surface area of cube = 6 * area of one face
    Surface area of cube = 6 * a2 
  2. Cuboid:

    In a cuboid length breadth and height are not equal.
    It has 6 square face,

    Volume of cuboid = length * breadth * height 
    
    Surface area of cuboid = 6 * area of one face
    Surface area of cuboid = length * breadth  
                              + length * breadth  
                              + breadth * height 
                              + breadth * height 
                              + length * height 
                              + length * height 
    
    Surface area of cuboid = 2(length * breadth  
                                  + breadth * height 
                                  + length * height ) 
  3. Sphere:

    A sphere has radius, volume and surface area:

    Volume of sphere = (4 / 3) * π * radius * radius * radius
    Volume of sphere = (4 / 3) * π * r3
    
    Surface area of sphere = 4 * &pi * radius * radius
    Surface area of sphere = 4 * &pi * radius2
    
    Volume of hemisphere = (2 / 3) * π * radius * radius * radius
    Surface area of sphere = 2 * &pi * radius * radius 
  4. Cylinder:

    A cylinder is a solid or hollow circular shape object with two circular base.

    Volume of cylinder = π * radius * radius * h
    Curved surface area of cylinder = 2 * π * radius * h
    Total surface area = 2 * π * radius * h + 2 * &pi * radius * radius 
  5. Cone:

    A cylinder is a solid or hollow with one circular base and which tappers from base to a point.

    Volume of a cone  = (1 / 3)π * radius * radius * height
    
    Curved surface area of a cone 
    = π * radius * slant height
    
    Total surface area of cone 
    = π * radius * slant height 
            + π * radius * radius 
  6. Pyramid:

    Pyramid is 3-d object with polygon base, this polygon base is connected to an apex in pyramid.

    Volume of a pyramid 
    = (area of base * height) / 3 

Examples:

  1. What will be the curved surface area of a cylinder having radius 5cm and height 10cm?
    Curved surface area of a cylinder 
    = 2 * π * r * height
    = 2 * 22 / 7 * 5 * 10 
    = 314.285714286 cm2 
  2. What will be the slant height of a cone which have curved surface area 990 cm2 and radius 5 cm?
    Curved surface area of a cone = π * radius * slant height 
    π * radius * slant height  = (22 / 7) * 5 * l
    l = 63cm 
  3. What is the volume of a square base pyramid having base side 4 cm and height 6cm?
    Volume of a square 
    = area of base * height / 3
    = 42 * 6 / 3
    = 16 * 6 / 3
    = 32 cm3 
  4. What will be the volume of a sphere having radius 2.1cm
    Volume of a sphere 
    = 4 * π * radius3 / 3
    = (4 / 3) * (22 / 7) * 2.13 
    = 38.808 cm3  
  5. What will be the surface area of a cube having side lenghth breadth and height 8 cm?
    Surface area of a cube 
    = 6 * side2
    = 6 * 82
    = 6 * 64
    = 384 cm2 


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