We develop an information geometric approach to symmetric cone programming. Information geometry is a differential geometric framework specifically tailored to deal with convexity arising in statistics, machine learning and signal processing, etc. In information geometry, Riemannian metric is defined as the Hessian of a convex potential function, and two mutually dual connections are introduced. We introduce an information geometric structure to convex programs by choosing the normal barrier function as the potential function. The two connections correspond to primal and dual problems. We focus on symmetric cone programs and demonstrate that the iteration-complexity of Mizuno-Todd-Ye predictor-corrector primal-dual path-following interior-point algorithm is expressed rigorously as an information geometric integral over the central path. Through extensive numerical experiments, we confirm that the iteration-complexity of the algorithm is explained quite well with the integral even for fairly large LP and SDP problem with thousands of variables. Endorsed by these numerical results, we claim that ``the number of iterations of the interior-point algorithm is a differential geometric quantity.''