A regular hexagon is formed by six wires each of resistance $r \Omega$ and the corners are joined to

Question

A regular hexagon is formed by six wires each of resistance $r \Omega$ and the corners are joined to the centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be

Professor PC Thomas Classes · Answer

This configuration represents a symmetrical bridge-like structure. By defining the input node as $A$ and the diametrically opposite output node as $D$, we can exploit the circuit's symmetry. Nodes symmetrically positioned relative to the $A-D$ axis will be equipotential, allowing us to effectively fold the circuit and reduce it into a combination of simpler parallel and series resistors. Evaluating this standard resistive mesh results exactly in an equivalent resistance of $R_{eq} = \frac{4}{5}r$.