Equivalent resistance between the adjacent corners of a regular n-sided polygon of uniform wire of resistance R would be:

$$\begin{gathered} \mathrm{Resistance} \mathrm{of} \mathrm{each} \mathrm{arm} \mathrm{is} \mathrm{R}/\mathrm{n} \mathrm{and} \\ \left(\mathrm{n}-1\right) \mathrm{such} \mathrm{resistances} \mathrm{are} \mathrm{in} \mathrm{series} \end{gathered}$$

$\mathrm{equivalent} \frac{\mathrm{R}}{\mathrm{n}}\left(\mathrm{n}-1\right)$

$\mathrm{with} \mathrm{this} \mathrm{one} \mathrm{R}/\mathrm{n} \mathrm{is} \mathrm{in} \mathrm{parallel}$

$R_e=\frac{(\mathrm{n}-1)\frac{\mathrm{R}}{\mathrm{n}}\cdot \frac{\mathrm{R}}{\mathrm{n}}}{(\mathrm{n}-1)\frac{\mathrm{R}}{\mathrm{n}}+\frac{\mathrm{R}}{\mathrm{n}}}=\frac{(\mathrm{n}-1)\mathrm{R}}{{\mathrm{n}}^2}$