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:

• Line 
  
• Triangle
  
• Quadrilateral
  
• Circle

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 

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 ) 

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 

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 

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 

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 

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 

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 

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  

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