Circles are too flat. What if we want to make a cylindrical Earth? We can take the same sort of approach, but we need a couple more parameters. Let’s say we have a cap radius R_c and height H_c, then we can make this:
With the sphere radius R_s, by matching the surface areas, the possible dimensions are restricted to 2 \pi R_c^2 + 2 \pi H_c = 4 \pi R_s^2
This one is easiest to do integrally. For the circular cap:
\cos(\phi) = 1-\frac{r^2}{2 R_s^2}
This defines the corner \phi in terms of the cap radius. Next, for the vertical portion, have z be the distance along the side of the cylinder. We find: 2 \pi \int_{\phi_c}^{\phi} R_s^2 \sin(t) dt = 2 \pi R_c z, yielding \cos(\phi) = 1- \frac{R_c (z+1/2)}{R_s^2}
And the bottom is basically like the top.