Shape description and feature detection are fundamental problems in computer graphics and geometric modeling. Among many existing techniques, those based on geodesic distance have proven effective in providing intrinsic and discriminative shape descriptors. In this article we introduce a new intrinsic function for a three-dimensional (3D) shape and use it for shape description and geometric feature detection. Specifically, we introduce the intrinsic girth function (IGF) defined on a 2D closed surface. For a point p on the surface, the value of the IGF at p is the length of the shortest nonzero geodesic path starting and ending at p. The IGF is invariant under isometry, insensitive to mesh tessellations, and robust to surface noise. We propose a fast method for computing the IGF and discuss its applications to shape retrieval and detecting tips, tubes, and plates that are constituent parts of 3D objects.