The cocoons surrounding powerful radio sources can be extensive if the
jet that feeds the cocoon is light and supersonic. They have been sho
wn to remain overpressured with respect to the ambient medium for most
of the lifetime of the sources. The observed lobes of the radio sourc
es form parts of these extensive cocoons. We show that observations of
the lobes of giant radio sources allow one to estimate the density of
the intergalactic medium (IGM) in which the lobes are embedded. We es
timate the IGM density to be of the order of a few per cent of the clo
sure density of the Universe. We further calculate the radio power of
the overpressured cocoon as a function of time and the ambient density
.