Gymnasium

A gymnasium (often called a gym) is a place used for physical activity. The word "γυμνάσιον" (gymnasion) was used in Ancient Greece. It meant a place for both physical and intellectual education of young men (see gymnasium (ancient Greece)). 
The intellectual education meaning persisted in German and other languages to mean a certain type of school providing secondary education, the Gymnasium, whereas in English the meaning of physical education was pertained in the word gym. However, the word "gymnasium" is not used very often in English nowadays. The term favoured in English is "sports centre", which perhaps seems more natural.
Some gyms offer memberships where people can train and improve their physical appearance. Commercial gyms offer sometimes help from qualified trainer. Again, another term is more used today: "fitness centre".
Sports such as basketball and volleyball are often played in gymnasia.