The question asked is a little vague. The radius of the circle in which the square mat an be placed such that it lies completely within the circle can take on an infinite number of values.
Let us assume you are referring to a circle that circumscribes the square mat and is one with the least area that can still accommodate the mat. The diameter of this circle is equal to the diagonal of the mat. As the mat is in the shape of a square with side 4 m, the length of the diagonal is sqrt(2*16) = 4*sqrt 2
The area of the circle is pi*((4*sqrt 2)/2)^2 = pi*4*2 = 8*pi
The area of the square is 4*4 = 16. This gives the area that is left uncovered by the mat as 8*pi - 16 = 9.132 m^2.
The area left uncovered when a square mat of side 4 m is placed in a circle that circumscribes it is is 9.132 m^2.