Ok, make it 14 instead of 10 then. That's a small correction. Filling the gaps on the other hand does not require a factor of two, but only 2 \pi / sqrt(27) or about 1.21. Multiplying the 14 with that gives us 17 instead of the 20 I estimated. I don't want to squabble about 20% here, but ask about the factor 100 discrepancy between naive estimate and projected number.
That is because you’re expecting ground access points to magically communicate with a satellite at 0 degree elevation. With a realistic elevation to deal with obstacles, diffraction, and electronic steering limitations, you’ll end up with a radius of coverage closer to 200 miles, which would give you that factor of 100. This shorter radius also limits losses due to distance attenuation. Your naive calculation assumes the limiting factor is the curvature of the only planet, which it is not; the earth could be a flat slab and you’d still need the same number of satellites.
