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).

Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.
Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain

The story

Edfu’s lots were donations to the temple and were irregular in shape. The scribes’ rule was practical and gave reasonable values ​​for almost rectangular terrain, but it showed a limit: without angles and diagonals, four sides do not determine a quadrilateral. The same four-bar frame can become a square or a flattened rhombus.

The Greeks solved the problem by dividing the quadrilateral by a diagonal into two triangles. In the 7th century, the Indian Brahmagupta found an exact formula for the area of ​​quadrilaterals that can be inscribed in a circle, using only the four sides.

In everyday life

  • Urban lots. Scriptures describe land on four sides and angles; the area is calculated by dividing the lot into triangles.
  • Gates and shelves. A quadrilateral of hinged bars deforms; a diagonal (like the “French hand”) stops it.
  • Kites. Many kites are quadrilaterals with two pairs of consecutive equal sides (deltoids).
  • Screens and sheets. A4 paper, cards and cell phone screens are quadrilaterals with right angles.

Sources: Elements, definition I.19 · Egyptian mathematics (MacTutor) · Brahmagupta (MacTutor).

Theory and constructions

DefiniçãoQuadrilateral

Let $A$, $B$, $C$ and $D$ be four points on the same plane, all distinct and three to three non-collinear. If the segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$ intersect only at the ends, their meeting is a quadrilateral:

$$ABCD = \overline{AB} \cup \overline{BC} \cup \overline{CD} \cup \overline{DA}.$$

The segments are the sides; $\hat{A} = D\hat{A}B$, $\hat{B} = A\hat{B}C$, $\hat{C} = B\hat{C}D$ and $\hat{D} = C\hat{D}A$ are the angles; $\overline{AC}$ and $\overline{BD}$ are the diagonals. The quadrilateral can be convex or concave.

A quadrilateral has 2 diagonals; the sum of the interior angles is $360^{\circ}$ (a diagonal divides it into two triangles) and the sum of the exterior angles is also $360^{\circ}$.