# Help with understanding waveguide dimensions for a pyramidal horn antenna

I want to build a pyramidal horn telescope as a science project but I’m having some trouble understanding the waveguide dimensions.

This documentation lists the dimensions of a paint thinner can be used (a = 16.8 cm and b = 10.48 cm). Although I understand the purpose for these dimensions it’s quite unclear to me why these values are used and how one could reach to their own custom values that would still work for the desired frequency of 1.42 GHz. Could any value be used that will yield the desired gain and cutoff frequency? Have I misunderstood something? Any help would be appreciated.

For directivity what primarily matters is the area of the open end of the horn antenna. If I assume the area of the open end of the horn is $$\ A \$$ then the directivity $$\ D \$$ is given as $$\ D = \dfrac{4\pi A}{\lambda^2} \$$. The directivity of the antenna is similar to gain but it does not account for reflections from the antenna or for losses in the antena.
Generally the gain of the antenna is given as $$\ G = e_r e_c D \$$. Here $$\ e_r \$$ is the reflection efficiency of the antenna. This accounts for the fact that part of the wave being fed into the antenna will be reflected back due to impedance mismatch. Meanwhile $$\ e_c \$$ accounts for the conductive losses in the antenna. $$\ e_c \$$ is more of a material constraint and something you probably cannot impact too significantly.