This is another tough nut and perhaps the diagram of the 50p
piece will help.
A 50 pence piece is a 7 sided
polygon ABCDEFG with rounded edges, obtained by replacing the
straight line DE with an arc centred at A and radius AE; replacing
the straight line EF with an arc centred at B radius BF
The 50p piece can roll in the same chute as a disc of radius $r$.
Suppose the seven arcs forming the edge of the 50p piece (the arcs
AB, BC etc. ) all have radius $R$ (where $R$=AD=AE=BE=BF...) then
you need to find $R$ in terms of $r$. These seven arcs subtend
angles of $2\pi /7$ at the centre of the disc and $2\pi /14$ at the