Carthage is located at (44.169550, -97.715801).
According to the United States Census Bureau, the city has a total area of 1.5 square miles (3.9 km²), of which, 1.5 square miles (3.8 km²) of it is land and 0.04 square miles (0.1 km²) of it (2.01%) is water.
There were 98 households out of which 14.3% had children under the age of 18 living with them, 49.0% were married couples living together, 4.1% had a female householder with no husband present, and 46.9% were non-families. 43.9% of all households were made up of individuals and 28.6% had someone living alone who was 65 years of age or older. The average household size was 1.91 and the average family size was 2.63.
In the city the population was spread out with 16.0% under the age of 18, 6.4% from 18 to 24, 16.0% from 25 to 44, 26.2% from 45 to 64, and 35.3% who were 65 years of age or older. The median age was 51 years. For every 100 females there were 98.9 males. For every 100 females age 18 and over, there were 84.7 males.
The median income for a household in the city was $27,679, and the median income for a family was $32,917. Males had a median income of $26,750 versus $15,938 for females. The per capita income for the city was $15,100. About 13.2% of families and 17.4% of the population were below the poverty line, including 40.0% of those under the age of eighteen and 12.1% of those sixty five or over.