Origins
A revolt is an overturning. The word arrived in English from French révolter, itself from Italian rivoltare — 'to turn upside down'. Behind both lies Latin revolvere: re- ('back') plus volvere ('to roll'). To revolt is to roll the existing order on its head.
The route through Italian is significant. English borrowed many political and military terms from Italian in the 16th century, when Italian city-states were laboratories of both governance and rebellion. Revolt, along with arsenal, infantry, and battalion, entered English with an Italian accent.
Latin volvere is one of the great productive roots. Revolve preserves the literal rolling motion. Revolution applies it to politics — a complete turn of the social wheel — and to astronomy, where planets make revolutions around their stars. Evolve means 'to roll out', as new forms unfold over time. Involve means 'to roll into', to become entangled.
Latin Roots
The most unexpected descendant is volume. A Roman volūmen was a rolled papyrus scroll. When books replaced scrolls, the name survived the change of format. A volume is still a roll, just in conceptual form.
The 'disgust' sense of revolt — 'that revolts me' — appeared by the 17th century, drawing on the idea of the stomach turning over. A revolting sight is one that rolls your insides.