Perhaps you are looking for a relationship in the form of (or wondering whether such a relationship exists):
Nmin(A_cap,n) = μ(n) * (A_Earth/A_cap)
A_cap: area of an instantaneous coverage by a single satellite, constant in time (circular orbits), modelled as a spherical cap.
Nmin: smallest number of satellites in a practical constellation that can provide seamless n-fold coverage.
μ(n): a fitting constant, independent of A_cap, function of n.
Fortunately, such approximate relationships exist, and μ(4)~7.2, whereas μ(1)~2.
However, I don’t know of any mathematical derivation of μ(n), even for μ(1). Mostly, μ(n) is derived from testing different constellations constructed using heuristic reasonings, such as the so-called Walker constellations (Wiki).
Let’s start with n=1 to get familiar with some published results.
Yuri Ulybyshev wrote a nice review in 2008 titled Satellite Design for continuous coverage: short historical survey
Figure 1, reproduced here for convenience, gives a plot for n=1 and Elevation =10° (that he called α).
As you have noted yourself, if you call Phi the Earth-centered half-cone angle of the spherical cap representing the individual coverage, then A_cap=2 π RE2 (1-cos(Phi))
So that (A_Earth/A_cap) = 2/(1-cos(Phi))
What I am claiming here is that the tight lower bound for N displayed in Figure 1 of Ulybyshev follows the trend of the relationship:
In other terms, μ(1) ~ 2.
Here is a check point so that we are on the same page with the detailed calculation:
H=1000Km (and El=10°) => Phi =21.6° => 2/(1-cos(Phi))= 28.4 => Nmin=56.8
This result (μ(1) ~ 2) was obtained independently by Beste in Design of satellite constellation for optimal continuous coverage. It is paywalled but Figure 3 is available (reproduced here, ψ is the half-cone angle that we called Phi).
Take GPS. Since we know H (20200 Km) we can compute their Phi (66.3°), assuming that their design Elevation is 10° (a reasonable assumption for Satnav). We also know GPS requires 24 satellites. From this we can make the informed guess that μ(4) ~ 7.2, assuming that the designers of GPS did optimize their constellation for minimum number of satellites.