The above rectangular piece of carpet has length 6 and width 4. if the largest possible circular hole is cut out of it, what is the area of the remaining piece?

The area of the remaining piece can be found by subtracting the area of the circle from the area of the rectangle.  
The diameter of the circle is the width of the rectangle. The radius is half of the diameter. Therefore the radius of the circle is 2. 
Area of the remaining piece = Area of the rectangular carpet - Area of the circular hole
 Area of the rectangular Carpet = length x width
  = 6 x 4 = 24 square units 
 Area of the circular hole = pi x radius x radius
  =3.14 x 2 x 2 = 12.56 square units
  Therefore, area of the rectangular carpet = 24 - 12.56 = 11.44 square units