Based on variable domain variational principle (VP), the variational theory of inverse shape design of heat conduction together with the boundary and the additional boundary conditions enforced on the unknown surface which must be satisfied by the problem is established. Incorporating the boundary and additional conditions into the functional makes the mathematic formulation very concise and compact. Through the established VP, both the temperature field and unknown boundary shape can be determined in a coupled way. The numerical method of the variational principle combined with finite element method is presented in the paper.