The Brownian plane is a basic model of random planar geometry, which appears as the scaling limit of discrete models such as the uniform infinite planar triangulation. We study the area of spheres centered at the distinguished point in the Brownian plane. As a function of the radius, the resulting process has continuously differentiable sample paths. Furthermore, the pair consisting of the process and its derivative is time-homogeneous Markov and satisfies an explicit stochastic differential equation.