curved surface area = total surface area ???

it is not necessarily correct.

you can think....
surface area has 2 different types
1) curved surface area
2) plane surface area

example one,
for a circular cone with base radius r, height h and slant height l
curved surface area = ( pi )( r )( l )
total surface area = curved surface area bsae area = ( pi )( r )( l ) ( pi )( r )^2
so, in this case, total surface area NOT = curved surface area

Example two,
for a sphere with radius r
as all the surface of sphere is curved,
total surface area = curved surface area = 4( pi )( r )^2
so, in this case, curved surface area = total surface area