'quadrilateral-based' is a modern English compound formed from 'quadrilateral' + 'based'. 'quadrilateral' derives from Latin elements 'quadri-' meaning 'four' and 'latus' meaning 'side', and 'based' is the adjective (past-participle) form of 'base' (from Latin 'basis' via Old French), meaning 'having a foundation'.

'quadrilateral' entered English via Medieval Latin 'quadrilateralis' (literally 'four-sided'), itself from Latin 'quadri-' + 'latus'. 'base' comes from Old French/Latin 'basis' and developed into the English noun and verb 'base', with the adjective/past participle 'based' used to form compounds like 'X-based'. The compound 'quadrilateral-based' is a recent formation using this pattern.

Initially, the components meant 'four' + 'side' and 'foundation' respectively; combined in modern usage they mean 'having or using a four-sided shape as the basis', a semantic composition rather than a major meaning shift.