For families, kinship coefficients are quantifications of the amount of genetic sharing between a pair of individuals. These coefficients are critical for understanding the breeding habits and genetic diversity of diploid populations. Historically, computations of the inbreeding coefficient were used to prohibit inbred marriages and prohibit breeding of some pairs of pedigree animals. Such prohibitions foster genetic diversity and help prevent recessive Mendelian disease at a population level. This paper gives the fastest known algorithms for computing the kinship coefficient of a set of individuals with a known pedigree. The algorithms given here consider the possibility that the founders of the known pedigree may themselves be inbred, and they compute the appropriate inbreeding-adjusted kinship coefficients. The exact kinship algorithm has running-time $O(n^2)$ for an $n$-individual pedigree. The recursive-cut exact kinship algorithm has running time $O(s^2m)$ where $s$ is the number of individuals in the largest segment of the pedigree and $m$ is the number of cuts. The approximate algorithm has running-time $O(n)$ for an $n$-individual pedigree on which to estimate the kinship coefficients of $\sqrt{n}$ individuals from $\sqrt{n}$ founder kinship coefficients.