  This paper revisits the optimal shape problem of a single cooling fin using a
one-dimensional heat conduction equation with convection boundary conditions.
Firstly, in contrast to previous works, we apply an approach using optimality
conditions based on requiring stationarity of the Lagrangian functional of the
optimisation problem. This yields an optimality condition basis for the
commonly touted constant temperature gradient condition. Secondly, we seek to
minimise the root temperature for a prescribed thermal power, rather than
maximising the heat transfer rate for a constant root temperature as previous
works. The optimal solution is shown to be fully equivalent for the two, which
may seem obvious but to our knowledge has not been shown directly before.
Lastly, it is shown that optimal cooling fins have a Biot number of 1,
exhibiting perfect balance between conductive and convective resistances.
