This article considers sample size determination for jointly testing a cause-specific hazard and the all-cause hazard for competing risks data. The cause-specific hazard and the all-cause hazard jointly characterize important study end points such as the disease-specific survival and overall survival, which are commonly used as coprimary end points in clinical trials. Specifically, we derive sample size calculation methods for 2-group comparisons based on an asymptotic chi-square joint test and a maximum joint test of the aforementioned quantities, taking into account censoring due to lost to follow-up as well as staggered entry and administrative censoring. We illustrate the application of the proposed methods using the Die Deutsche Diabetes Dialyse Studies clinical trial. An R package "powerCompRisk" has been developed and made available at the CRAN R library.