Area of the planet's atmosphere 2 pi Rp h
----------------------------------- = ---------------
Area of entire photosphere pi Rs*Rs
Rp h
= 2 ---- * ---
Rs Rs
Here, "h" is something proportional to the scale height
of the planet's atmosphere. I'll use a very rough value
which is probably on the large side.
For our (cold) Jupiter, the scale height is about 30 km,
so I'll choose "h" = 300 km for a (hot) Jupiter.
For a typical hot Jupiter around a Sun-like star, the ratio is
roughly
9 x 10^(-5) ~ 10^(-4)
For our Earth, the scale height is very roughly 8 km,
so I'll choose "h" = 8 km.
That means for an Earth-sized planet around a Sun-like star, the ratio is
roughly
2 x 10^(-7)