In the previous post I supposed a torus roughly the radius of Earth, but because of my arbitrarily chosen proportions, the torus had only a fraction of Earth's surface area and volume. In this post I'll work out the dimensions that would give a torus a surface area and volume closer to Earth's.
The surface areas and volumes of Earth and a torus are
Setting the surface areas equal to each other, and the volumes equal to each other,
gives us two equations with two unknowns, R and r. Let's solve the first equation for R:
Plug it into the second and solve for r:
Now plug that back into the equation for R:
Our final equations, then, for the torus's major and minor radii are
To have the same surface area and volume as Earth, our planetary toroid would have to have a major radius about five times larger than Earth and a minor radius about a fifth of Earth's. Compared to the drawing in the previous post, such a torus would look more like this (Earth's size shown at the center):
While this could be an interesting scenario to explore, I'd like to stick with a smaller, fatter torus, which I think will provide some more interesting skyscapes.