  The Glauber model analysis of the elastic $^{6,8}$He+$p$ scattering data at
energies around 700 MeV/nucleon, measured in two separate experiments at
GSI-Darmstadt, has been done using several phenomenological parametrizations of
the nuclear matter density. By taking into account the new data points measured
at the high momentum transfer, the nuclear matter radii of $^{6,8}$He were
accurately determined from the Glauber model analysis of the data, with the
spin-orbital interaction explicitly taken into account. The well-known geometry
for the core and dineutron halo has been used with the new parametrizations of
the $^{6}$He density to extract the detailed information on the structure of
$^{6}$He in terms of the core and dineutron halo radii. An enhanced sensitivity
of the data measured at the high momentum transfer to the core part of the
$^{6,8}$He densities has been found.
