You can start with simpler formula:
L ≈ 0.45 * λ
20,
0.45 - empirical coefficient mainly for open end effect
λ
20 - wavelength for 20 ohm microstrip line (wavelength in wide patch seems to be similar to wide low-impedance microstrip line)
Then use L for both patch width and patch height.
For example:
F=10.525 GHz
Er=4.3
t=1[mm]
20 ohm microstrip line will have λ
20≈14.4
Then L ≈ 0.45*14.4 = 6.48 [mm]
Patch height = patch width = 6.48 [mm]
Remaining part is to obtain patch input impedance Z
in by some formula or by simulation.
some more formulas here:
https://www.antenna-theory.com/antennas/patches/patch3.php
update: I used different formulas to calculate Z
in, found in different papers. Results was very different, around 100 to 300 Ohm for the same patch W/H , Er, substrate thickness. So do not rely on those formulas for too much.