Trapezium

Areas of flat figures

How many square meters are there on land, how much paint goes on a wall, how much fabric goes on a flag? All these questions are about area — and almost all the formulas come from just one: the rectangle. Who discovered The first area calculations that we have records of are from scribes from Egypt and Mesopotamia: the Rhind papyrus (c. 1650 BC) calculates areas of rectangular, triangular and even circular fields, and Babylonian tablets show areas of trapezoids. The Greeks gave these rules a justification: in The Elements, Euclid (c. 300 BC) proves that parallelograms with the same base and the same height have the same area (proposition I.35) and that the triangle is half of the parallelogram (I.41). The circle formula was demonstrated by Archimedes (3rd century BC).

October 12, 2026 Read

Exercises from chapter 7 — Notable quadrilaterals

Exercises and resolutions 224 Determine the value of $x$ in the following cases: Ver respostaa) $x = 120^{\circ}$ · b) $x = 75^{\circ}$ 225 Determine the angles of the quadrilateral $ABCD$ in the cases: Ver respostaa) $130^{\circ}$, $70^{\circ}$, $95^{\circ}$, $65^{\circ}$ · b) $55^{\circ}$, $105^{\circ}$, $70^{\circ}$, $130^{\circ}$ 226 Determine the value of $x$ in the cases: a) $PA = PB$; b) $AB = AD$ and $CB = CD$. Ver respostaa) $x = 35^{\circ}$ · b) $x = 70^{\circ}$

October 11, 2026 Read

Consequences — medium bases

Connect the midpoints of two sides of a triangle: the segment is parallel to the third side and measures half of it. Who discovered The mean base theorem is a particular case of proposition VI.2 of The Elements of Euclid: a straight line parallel to one side of a triangle divides the other two sides in the same proportion. A beautiful consequence has a well-known author: the Frenchman Pierre Varignon (1654–1722) showed that, connecting the midpoints of the sides of any quadrilateral, a parallelogram is always obtained. The result was published after his death, in 1731.

October 11, 2026 Read

Rectangle, rhombus and square

Diagonals tell a lot about a quadrilateral. In the rectangle, they are equal; in the rhombus, perpendicular; in the square, both. Who discovered The most famous document on the diagonal of the square is the Babylonian tablet YBC 7289, from around 1800–1600 BC, today at Yale University. It shows a square with diagonals and a value of $\sqrt{2}$ with an error of less than one millionth. The properties of the diagonals of the rectangle and the rhombus follow from the theorems on parallelograms of The Elements of Euclid.

October 11, 2026 Read

Properties of parallelograms

Opposite sides equal, opposite angles equal, diagonals that cut in half. The parallelogram is the figure of forces and articulated mechanisms. Who discovered The first properties are in the proposition I.34 of the The Elements of Euclid (c. 300 BC): in a parallelogram, the opposite sides and angles are equal, and the diagonal divides it into two parts of the same area. Outside of geometry, the parallelogram gained fame with the parallelogram rule of forces, used by the Flemish Simon Stevin (1586) and enunciated by Isaac Newton in the Principia (1687): two forces acting at a point are equivalent to the diagonal of the parallelogram formed by them.

October 11, 2026 Read

Properties of trapezoids

The trapezoid has a pair of parallel sides — and is the shape hidden in many ramps, rooftops, and pockets. Who discovered The modern meaning of the word comes from Posidonius (1st century BC) and Proclus (5th century AD): he records that Posidonius separated quadrilaterals with a pair of parallel sides from those that had none. For Euclid, trapezoid was any quadrilateral that was not a square, rectangle, rhombus or parallelogram. The area of the trapezoid — the average of the bases times the height — appears in the works of Heron of Alexandria (1st century AD).

October 11, 2026 Read

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.

October 11, 2026 Read

Quadrilateral — definition and elements

Four sides, four vertices, two diagonals. Quadrilaterals are in lots, screens, tables and windows. Who discovered Calculating the area of land on four sides was the task of the scribes of Egypt and Mesopotamia. Inscriptions from the temple of Edfu, in Egypt (2nd century BC), record a rule for the area of a four-sided plot: multiply the average of two opposing sides by the average of the other two. The rule is only exact for rectangles and overestimates the area in other cases. The geometric definition — figure limited by four straight lines — is from Euclid (definition I.19 of The Elements).

October 11, 2026 Read