Cell size is specific to each species and impacts cell function. Various phenomenological models for cell size regulation have been proposed, but recent work in bacteria has suggested an 'adder' model, in which a cell increments its size by a constant amount between each division. However, the coupling between cell size, shape and constriction remains poorly understood. Here, we investigate size control and the cell cycle dependence of bacterial growth using multigenerational cell growth and shape data for single Caulobacter crescentus cells. Our analysis reveals a biphasic mode of growth: a relative timer phase before constriction where cell growth is correlated to its initial size, followed by a pure adder phase during constriction. Cell wall labelling measurements reinforce this biphasic model, in which a crossover from uniform lateral growth to localized septal growth is observed. We present a mathematical model that quantitatively explains this biphasic 'mixer' model for cell size control.