We derive the quasiconvex relaxation of the Biot-type energy density ∥sqrt(Dφ^TDφ)−I_2∥^2 for planar mappings φ:R2→R2 in two different scenarios. First, we consider the case Dφ∈GL+(2), in which the energy can be expressed as the squared Euclidean distance dist2(Dφ,SO(2)) to the special orthogonal group SO(2). We then allow for planar mappings with arbitrary Dφ∈R2×2; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks.