Effective resistance of a cut toroid
You shall cut into slices along the circumference.

Each slice will be \$\text{d}r\$ wide (from \$r\$ to \$r+\text{d}r\$), \$W\$ tall and \$2\pi r\$ long neglecting the radial cut.

Since all those slices will linearly drop the same voltage along the same angular distance we can assume there will be no radial currents as confirmed by this Femm4.2 simulation of a solid copper slotted toroid fed by 1V.



Above volatge plot shows no radial voltage gradient, i.e. no radial electric field nor current density.

Hence all those slices will simply be parallel connected, just like in a kind of array of balanced Wheatstone bridges where "horizontal" branches carry no current at all.







simulate this circuit &ndash; Schematic created using CircuitLab

Discrete elements schematics

In this case it's much simpler summing individual conductivities instead of paralleling resistances.

Each slice would have resistance \$R=\rho\frac{l}{S}\$ and so \$G=\sigma\,\frac{S}{l}\$ in this case

$$ \text{d}G=\sigma\,\frac{W}{2\pi r}\,\text{d}r\quad\Rightarrow\quad G=\sigma\,\frac{W}{2\pi}\int_{r_i}^{r_o}\frac{\text{d}r}{r}=\sigma\,\frac{W}{2\pi}\,\ln\frac{r_o}{r_i}$$

Switching to resistance we get

$$ R=\rho\,\frac{2\pi}{W\ln\frac{r_o}{r_i}} $$

Interesting enough if \$r_o \approx r_i\$ the above tends to

$$R=\rho\,\frac{2\pi}{W\ln\frac{r_o}{r_i}}\rightarrow \rho\,\frac{2\pi\,\frac{r_o+r_i}{2}}{W(r_o-r_i)}=\rho\,\frac{l}{W\Delta r}$$

...tends to what first reckoned out using ring width and average length.

It may finally prove interesting to find out current density distribution.

Again each slice can be studied on its own so

$$ I=VG\quad \text{d}I=V\text{d}G=V\sigma\frac{W}{2\pi r}\text{d}r=V\sigma\frac{\text{d}S}{2\pi r}$$

being \$\text{d}S=W\text{d}r\$ the infinitesimal cross section of the slice 

$$J=\frac{\text{d}I}{\text{d}S}=\frac{\sigma}{2\pi r}V $$

In short we have an hyperbolic current density distibution against the radius. This too is quite consistent: inner radius, shorter path, higher current.



Current density plot and J(r) along red radius on top of toroid

And again confirmed by the finite elements simulation.