Origins
The word 'hyperbole' entered English in the fifteenth century from Latin 'hyperbole,' borrowed directly from Greek 'hyperbolḗ' (ὑπερβολή), meaning 'excess,' 'exaggeration,' or literally 'a throwing beyond.' The Greek word is composed of 'hypér' (ὑπέρ, over, beyond) and 'bolḗ' (a throw, a casting), from the verb 'bállein' (to throw). A hyperbole is, at its etymological root, an overthrow — a statement hurled past the boundary of truth for rhetorical effect.
The Greek verb 'bállein' (to throw) comes from PIE *gʷelh₁- (to throw) and has been extraordinarily productive in English through Greek borrowings. 'Ballistic' (pertaining to thrown projectiles), 'ball' (a thrown object — though also possibly from Old Norse), 'parable' (a throwing beside — a comparison, a story placed alongside reality), 'symbol' (a throwing together — a sign that represents something), 'problem' (a thing thrown forward — an obstacle set before you), 'metabolism' (a throwing across — a change, a transformation), 'devil' (from Greek 'diábolos,' a slanderer, literally 'one who throws across' — one who casts accusations), and 'diabolical' all trace back to 'bállein.'
The prefix 'hypér' (over, beyond) comes from PIE *upér, which also produced Latin 'super' and English 'over.' The correspondence hyper/super/over is one of comparative linguistics' clearest demonstrations: the same PIE word took different phonological paths in Greek (hyper-), Latin (super-), and Germanic (over-), but all three preserve the meaning 'above' or 'beyond.'
Latin Roots
As a rhetorical figure, hyperbole was catalogued and discussed by the ancient Greek and Roman rhetoricians. Aristotle mentioned it in the 'Rhetoric.' Quintilian discussed it extensively in the 'Institutio Oratoria,' defining it as 'an elegant straining of the truth.' The figure's effectiveness depends on the audience understanding that the statement is not literal: 'I have told you a million times,' 'I am so hungry I could eat a horse,' 'this bag weighs a ton.' The exaggeration is the point — it communicates the speaker's intensity of feeling rather than a factual claim.
The mathematical term 'hyperbola' (a conic section) uses the identical Greek word. Apollonius of Perga, the Greek mathematician who classified the conic sections around 200 BCE, named three of them using throwing metaphors: 'ellipse' (a falling short — the section 'falls short' of the cone's side), 'parabola' (a placing beside — the section runs parallel to the side), and 'hyperbola' (a throwing beyond — the section 'exceeds' the side). The mathematical and rhetorical meanings thus share not just a root but the same concept: going beyond a standard.
The distinction between 'hyperbole' (the rhetorical figure, stressed on the second syllable: hy-PER-bo-lee) and 'hyperbola' (the mathematical curve, stressed on the second syllable: hy-PER-bo-la) is maintained in modern English, though both derive from the same Greek noun. The rhetorical term preserves the Greek feminine ending '-ē,' while the mathematical term uses the Latinized ending '-a.'
Legacy
Hyperbole is sometimes confused with lying, but they are fundamentally different. A lie intends to deceive; a hyperbole intends to express. The statement 'I waited for ages' does not deceive anyone — both speaker and listener understand the exaggeration. Hyperbole is a shared convention, a form of emotional emphasis that paradoxically communicates truth (about the speaker's feelings) through falsehood (about the facts).