The floor of the family room is an area, with a length and width. The measurement of an area would be __^2, in this case m^2.
Additionally, you can think about it in the form of multiplying your units:
6 m in length x 4 m in width is the area
6 m x 4 m = 6 x 4 x m x m = 24m^2
In this case, the unit is m^2.
