The surface area of an $n$-dimensional ellipsoid is expressed in terms of hyperelliptic integrals, see <a href="http://nzjm.math.auckland.ac.nz/images/b/b3/Surface_area_and_capacity_of_ellipsoids_in_n_dimensions.pdf">SURFACE AREA AND CAPACITY OF ELLIPSOIDS IN n DIMENSIONS</a> by Garry J. Tee When $n=3$ they are elliptic integrals, and the result is due to Legendre. The paper also mentions approximate formulas.