Managing rapid population growth and spurring economic growth are among the most pressing policy challenges for Sub-Saharan Africa. We discuss the links between them and investigate the potential of family planning programs to address these challenges. Specifically, we estimate the impact of family planning programs on income per capita that can arise via the demographic dividend (DD), a boost to per capita income that operates through a chain of causality related to declining fertility. We develop a model to determine the impact of "meeting unmet need" (MUN) for modern contraceptive methods on fertility and hence on the population age structure in the coming years. We also estimate empirically the DD that has been observed in other countries, using a cross-country regression with panel data covering 40 years. Using the age structure projected by MUN and the empirical estimates of the DD, we estimate the potential for additional economic growth in Kenya, Nigeria, and Senegal. We find that in 2030, these countries can enjoy an increase in per capita income of 8-13% by meeting one-third of their unmet need for modern contraception and can enjoy a 31-65% higher income per capita by meeting all of the unmet need. By 2050, these ranges become 13-22% and 47-87% respectively. We discuss the policy implications of our findings.