The township is named after Thomas Cutts of Maine, who received a land grant from the New Hampshire state legislature in 1810.

[citation needed] According to the United States Census Bureau, the grant has a total area of 11.3 square miles (29.3 km2), all land other than streams.

The east and west boundaries roughly parallel the river, each approximately one mile distant.

The Dry River enters the valley of the Saco River in Crawford Notch at the southern end of the township, and the northern end is below Oakes Gulf on the southern slopes of Mount Washington.

The highest point in Cutt's Grant is its northeastern corner, located just south of Gulf Peak on a southeast spur of Mount Washington, where the elevation reaches 4,720 feet (1,440 m) above sea level.