/ AbstractWe present 3D calculations for dielectric haloscopes such as the currently envisioned MADMAX experiment. For ideal systems with perfectly flat, parallel and isotropic dielectric disks of finite diameter, we find that a geometrical form factor reduces the emitted power by up to $30\,\%$ compared to earlier 1D calculations. We derive the emitted beam shape, which is important for antenna design. We show that realistic dark matter axion velocities of $10^{-3} c$ and inhomogeneities of the external magnetic field at the scale of $10\,\%$ have negligible impact on the sensitivity of MADMAX. We investigate design requirements for which the emitted power changes by less than $20\,\%$ for a benchmark boost factor with a bandwidth of $50\,{\rm MHz}$ at $22\,{\rm GHz}$, corresponding to an axion mass of $90\,μ{\rm eV}$. We find that the maximum allowed disk tilt is $100\,μ{\rm m}$ divided by the disk diameter, the required disk planarity is $20\,μ{\rm m}$ (min-to-max) or better, and the maximum allowed surface roughness is $100\,μ{\rm m}$ (min-to-max). We show how using tiled dielectric disks glued together from multiple smaller patches can affect the beam shape and antenna coupling.