Origins
The word 'rectangle' entered English in 1571 from Late Latin 'rectangulum,' a compound of 'rēctus' (right, straight) and 'angulus' (angle, corner). The word means, with transparent literalness, 'right-angled' — a shape defined by having all four of its interior angles at exactly 90 degrees.
The first element, 'rēctus,' is the past participle of 'regere' (to guide, to make straight), from PIE *h₃reǵ- (to move in a straight line). This root connects 'rectangle' to the vast family of 'straightness' words: 'correct,' 'erect,' 'direct,' 'regime,' 'regal,' 'right,' and 'rule.' A 'right angle' is a 'straight angle' — the angle that a plumb line makes with the ground, the angle of perfect perpendicularity. The word 'right' itself (meaning correct, just, and the opposite of left) descends from the same PIE root through the Germanic branch.
The second element, 'angulus' (angle, corner), comes from PIE *h₂enk- (to bend). This root also produced English 'ankle' (the bend of the foot), 'anchor' (a bent hook), and Greek 'ankýlos' (crooked, bent — source of 'ankylosis,' a stiffening of a joint). The Indo-European mind conceived an angle as a bend — a point where a straight line changes direction.
Development
The combination is etymologically paradoxical: a rectangle is a 'straight bend,' a shape where the concepts of straightness and bending coexist. The paradox resolves geometrically: at a right angle, two straight lines meet in the most balanced possible way — neither acute (too sharp) nor obtuse (too blunt), but exactly perpendicular. The 'rightness' of a right angle lies in its perfect balance.
The Greeks and Romans had different terms for what we call a rectangle. Greek mathematicians used 'orthogṓnion' (ὀρθογώνιον), from 'orthós' (straight, right) and 'gōnía' (angle) — the same compound concept, rendered in Greek rather than Latin elements. The Latin term 'rectangulum' was a later coinage, probably from the medieval period, when Latin mathematical vocabulary expanded to meet educational needs.
In Euclid's Elements (c. 300 BCE), the properties of rectangles are derived from more basic postulates and propositions. Euclid proved that opposite sides of a rectangle are equal and parallel, that its diagonals bisect each other, and that it is a special case of a parallelogram. The rectangle — along with the triangle and the circle — became one of the foundational shapes of Western geometry.
Scientific Usage
The special case where all four sides are equal produces a square, which is technically a rectangle (all angles are right angles) but is distinguished by its additional symmetry. The relationship between rectangle and square — where the square is a subcategory of the rectangle — has been used in mathematics education as an example of category inclusion, often confusing students who feel that a square should not 'count' as a rectangle.
Modern descendants include 'rectangular' (the adjective form), 'rectilinear' (moving in straight lines), and 'rectify' (to make right or straight). The mathematical constant called the 'rectification' of a curve refers to finding the curve's length — literally 'straightening' it. In each case, the ancient PIE equation of straightness with correctness persists: to be rectangular is to be right-angled, and to be right-angled is to be geometrically correct.