What Roman underfloor heating system used hot air from a furnace?

The Roman underfloor heating system that used hot air from a furnace was the hypocaust.

A hypocaust raised a room’s floor above the ground on small brick or tile pillars called pilae stacks. Hot air and smoke from a wood-burning furnace, or praefurnium, flowed through the enclosed space beneath the floor. In more elaborate buildings, the heat also passed through hollow wall tiles and flues, warming the walls before escaping through roof outlets.

Hypocausts were especially important in Roman public baths, where they heated the hottest rooms, including the caldarium, and sometimes helped warm water boilers. They also appeared in villas and other substantial buildings across the Roman Empire. Because the system consumed large quantities of fuel and required constant attention to the fire, it was generally too costly for ordinary homes.

The other options describe different architectural features. A peristyle is a colonnade around a courtyard, an atrium is a central hall or courtyard space, and an aqueduct carries water. Hypocaust technology is an ancestor of modern radiant or underfloor heating, although it moved hot air rather than circulating heated water through pipes.