Surface Area of Ellipsoids

We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity properties of the surface area, to give sharp estimates for the surface area of a large-dimensional ellipsoid, to produce asymptotic formulas in large dimensions, and to give an expression for the surface in terms of the Lauricella hypergeometric function.