So far I've calculated the effective aperture using an equation. However I was wondering how to calculate dish diameter using the gain calculated earlier and if there's an equation for this?
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If you know effective area, you can calculate the physical area by dividing by efficiency.
Typical efficiency for a dish is about 60%, possibly less at low frequencies (under 1 GHz) or if it's a simple home made feed (yagi or straight waveguide). 70% is achievable with a choke, 80% in very special cases with a lot of optimising.
So $A_{dish} = A_e / 0.6$
From physical area you can work out diameter with $A_{dish}=\frac\pi4 D^2$
Efficiency here is not a true loss of power, it's a measure of how the dish area is not used perfectly. In transmitting, the dish is not illuminated uniformly, so the field is not uniform over the aperture. This is a natural consequence of balancing under-illumination (wasted area on the sides) with over-illumination (wasted power spilling over). The feed has a gradual taper, it can't just stop at the very edge. Also a tapering of the dish illumination gives a cleaner shape beam with lower sidelobes, if this matters to you.
