Inductance of a straight vs toroidal solenoid

[htg]

According to widely known formulas, the inductance of a long straight solenoid will remain practically the same if we bend it into a toroidal solenoid. It seems to be a contradiction, because the magnetic path length of a straight solenoid is more than two times greater.

 

[AJ Bentley]

The path length isn't actually important. What matters is the strength of the magnetic field.

The long solenoid approximation is used because it allows a simplifying assumption that you can ignore any field outside the solenoid as trivially small when you calculate the field inside. (using Stokes theorem and curl B = mu0 J)

In the case of a toroid, the field actually IS zero outside so the assumption becomes a fact.
The difference between the two is trivial.

 

[htg]

I find it strange (in the light of the Gauss' law) that the flux going through the inside of a long solenoid just comes back on the outside no matter how long the solenoid is.

In the case of a toroid, I don't think the field on the outside is zero - it is not confined to such a small volume, so its intensity is much smaller.