We designed three-dimensional electromagnetic cloaks, starting from a small region of complex shape instead of a point. We derive the expression of a transformation matrix describing an object with a surface of revolution and its associated non-singular cloak. We note that while none of the eigenvalues vanish inside the cloak, they suffer a discontinuity on its inner surface. Moreover, all three eigenvalues are independent of the radius in the concealed object. The validity of our analytical results is confirmed by finite edge-element computations showing scattering is much reduced when two objects are dressed with their accompanying cloak. We note that neither the objects nor the cloaks are invisible on their own.