The 10j symbol is a spin network that appears in the partition function for the Barrett–Crane model of Riemannian quantum gravity. Elementary methods of calculating the 10j symbol require O(j⁹) or more operations and O(j²) or more space, where j is the average spin. We present an algorithm that computes the 10j symbol using O(j⁵) operations and O(j²) space, and a variant that uses O(j⁶) operations and a constant amount of space. An implementation has been made available on the web.