  Given a Riemmanian manifold, we provide a new method to compute a sharp upper
bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a
geodesic ball of radius less than the injectivity radius of the manifold. This
upper bound is obtained by transforming the metric tensor into a rotationally
symmetric metric tensor that preserves the area of the geodesic spheres. The
provided upper bound can be computed using only the area function of the
geodesic spheres contained in the geodesic ball and it is sharp in the sense
that the first eigenvalue of geodesic ball coincides with our upper bound if
and only if the mean curvature pointed inward of each geodesic sphere is a
radial function.
