This study introduces a novel constraint in linkage mechanism design: the constant-area constraint. Traditional linkage mechanisms rely on constraints of rigidity and joint motion, the constant-area constraint enforces an additional limitation:  the area enclosed by the linkages remains constant throughout its motion. Originating from incompressible fluid mechanics, this concept is adapted for linkage mechanisms. We present computational tools developed for analysing and optimising constant-area linkage mechanisms. These tools are applied to explore their kinematic behaviour and possibilities. This novel concept led to a modification of the Gruebler degrees-of-freedom formula. The constant area concept is applied to five- and six-bar mechanisms and an exploratory analysis of multiple interconnected constant area cells is also conducted.