  In this paper we revisit Brill's proof of positive mass for
three-dimensional, time-symmetric, axisymmetric initial data and generalise his
argument in various directions. In 3+1 dimensions, we include an apparent
horizon in the initial data and prove the Riemannian Penrose inequality in a
wide number of cases by an elementary argument. In the case of 4+1 dimensions
we obtain the analogue of Brill's formula for initial data admitting a
generalised form of axisymmetry. Including an apparent horizon in the initial
data, the Riemannian Penrose inequality is again proved for a large class of
cases. The results may have applications in numerical relativity.
