Fairplay Township is one of fifteen townships in Greene County, Indiana, United States. As of the 2020 census, its population was 505, down from 575 at 2010.
According to the 2010 census, the township has a total area of , of which (or 98.70%) is land and (or 1.26%) is water. The stream of Lattas Creek runs through this township.
(This list is based on USGS data and may include former settlements.)
The township contains one cemetery, Castle Hill.