Humans appear to intuitively grasp definitions foundational to formal geometry, like definitions that describe points as infinitely small and lines as infinitely long. Nevertheless, previous studies exploring human's intuitive natural geometry have consistently focused on geometric principles in planar Euclidean contexts and thus may not comprehensively characterize humans' capacity for geometric reasoning. The present study explores whether children and adults can reason about linearity in spherical contexts. We showed 48 children (age range: 6-8 years) and 48 adults from the U.S. Northeast two different paths between the same two points on pictures of spheres and asked them to judge which path was the most efficient for an actor to get from a starting point to a goal object. In one kind of trial, both paths looked curved in the pictures, and in another kind of trial, the correct curved-looking path was paired with an incorrect straight-looking path. Adults were successful on both kinds of trials, and although children often chose the incorrect straight-looking path, they were surprisingly successful at identifying the efficient path when comparing two that were curved. Children thus may build on a natural geometry that gives us humans intuitions that are not limited to the formal axioms of Euclidean geometry or even to the Euclidean plane.