We prove the uniqueness of a solution to the mean curvature equation for graphs for any radial continuous initial data. The existence of a smooth solution to this problem comes from the work of Ecker and Huisken. We obtain in addition that the solution is radial. We point out that existence and uniqueness hold without any growth restriction on the solution or the initial data, a situation which is rather different to the related stationary problem: in this case, we show there is a limiting growth above which uniqueness does not hold anymore. An application of the uniqueness result to the evolution by mean curvature of entire radial graphs is given.