With smooth objective functions and constraint conditions, gradient-based methods can be used to solve multi-objective optimization problems efficiently. However, when applied to structural sizing optimization problems, using the Finite Element Method (FEM) and a finite difference scheme to calculate sensitivities can be computationally expensive. The adjoint variable method can be used to reduce computational cost. In order to solve multi-objective structural sizing and shape optimization problems efficiently, this paper proposes using the adjoint variable method. The adjoint variable method efficiently calculates multiple sensitivities for objectives that involve structural responses and cuts down computational cost by reducing the number of sensitivity calculations required per design variable.