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GRE Geometry | Three – Dimensional Figures

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  • Last Updated : 20 Jan, 2022

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: 
     

  1. In a cube length breadth and height are all equal. 
     
length = breadth = height 
  1. 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 
  1.  
  2. Cuboid: 
     

  1. 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 ) 
  1.  
  2. Sphere: 
     

  1. 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 
  1.  
  2. Cylinder: 
     

  1. 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 
  1.  
  2. Cone: 
     

  1. 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 
  1.  
  2. Pyramid: 
     

  1. 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 
  1.  

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 
  1.  
  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 
  1.  
  2. 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 
  1.  
  2. 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  
  1.  
  2. What will be the surface area of a cube having side length breadth and height 8 cm? 
     
Surface area of a cube 
= 6 * side2
= 6 * 82
= 6 * 64
= 384 cm2 
  1.  

 

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