This article proposes a new, fast method to solve inverse acoustic problems for planar sources. This problem is often encountered in practice and methods such as planar nearfield acoustic holography (PNAH) and statistically optimised nearfield acoustic holography (SONAH) are widely used to solve it. In this article, it is shown that the transfer matrix from an equidistant source grid to an equidistant field grid has a Block Toeplitz with Toeplitz Blocks (BTTB) structure. By using fast methods for BTTB matrices, the computational effort is reduced to a similar level as PNAH at an accuracy which is the same as SONAH. This is demonstrated in a case study, where computational time and the accuracy is compared. The accuracy is shown to be better than PNAH even if advanced sound field extrapolation techniques such as patch holography are used.