Euler's Eyes
2026-10-11
Notable quadrilaterals — definitions
Trapezoid, parallelogram, rectangle, rhombus and square. The names are old, but what each means has changed over the centuries.
Who discovered
Squares and rectangles already appear in Babylonian tablets and in Egyptian area problems. The definitions came from Euclid (definition I.22 of The Elements), who distinguished square, oblong, rhomboid and rhomboid and called trapezoids all other quadrilaterals. The word parallelogram appears for the first time in Euclid’s proposition I.34. The inclusive classification used today, in which every square is a rectangle, was adopted at the end of the 19th century, as explained in the curiosity “Definitions changed over time”, below.
The story
The names contain images: trapezium comes from the Greek trapezion, “little table”; rombo, from rhombos, a top that spun making noise; lozenge comes from the French losange, linked to heraldry, the art of coats of arms.
For centuries, the same name has meant different things in different places. To this day, the trapezium in British English is the trapezoid in American English, and vice versa — a change attributed to the mathematics dictionary of Charles Hutton, from 1795.
In everyday life
- Warning signs. In Brazil, yellow warning signs are squares supported on a vertex, shaped like a diamond.
- Floors and tiles. Rectangles and squares fill the plan without leaving gaps.
- Spantograph scissors. Freight elevators, folding gates and articulated luminaires use parallelograms that deform without losing parallelism.
- Screens. The 16:9 format of TVs and the A4 size of paper are rectangles with standardized proportions.
Sources: Elements, definition I.22 · Elements, proposition I.34 · Charles Hutton (MacTutor).
Theory and constructions
DefiniçãoThe five notable quadrilaterals
They are all flat convex quadrilaterals:
- Trapezium: has two parallel sides (the bases). It is isosceles if the non-base sides are congruent, scalene if they are not, and rectangle (or birectangle) if it has two right angles.
- Parallelogram: has parallel opposite sides: $\overline{AB} \parallel \overline{CD}$ and $\overline{AD} \parallel \overline{BC}$.
- Rectangle: has four congruent angles.
- Rhombus: has four congruent sides.
- Square: it has four congruent angles and four congruent sides.
