Your thinking works out correctly for the area of the ellipse and the volume of the ellipsoid.
For the perimeter of the ellipse, your short formula is a rough approximation to the true value. This link has a good discussion. It gives the very good approximation as
where The short formula drops all the terms with in them.
The link gives an example comparing the accuracy. When and the short formula gives 65.9734. The above approximation is 69.039336580. The true value is 69.03933778699452855....