We 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 $90,mu{rm eV}$. We find that the maximum allowed disk tilt is $100,mu{rm m}$ divided by the disk diameter, the required disk planarity is $20,mu{rm m}$ (min-to-max) or better, and the maximum allowed surface roughness is $100,mu{rm m}$ (min-to-max). We show how using tiled dielectric disks glued together from multiple smaller patches can affect the beam shape and antenna coupling.
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