All this manual calculation rather than just invoking the isoperimetric inequality? I mean, manual calculation isn't bad, if you want to do it also as an additional demonstration, but I think the isoperimetric inequality is worth a mention here!

The isoperimetric inequality is correct but too short. You need a longer proof with more equations and integrals to convince people.

The calculation is somewhat bogus since comparing a sphere and a torus with the same radius isn't really valid.

The author should either compare the surface areas of spheres and toroi with the same volume, or vice versa.