Propositiones ad Acuendos Juvenes/Problems/23 - De Campo Quadrangulo/Historical Note

Historical Note on Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $23$: De Campo Quadrangulo

The formula being used here for a quadrilateral whose sides are $a$, $b$, $c$ and $d$ is the Egyptian Formula for Area of Quadrilateral:

$\AA = \dfrac {a + b} 2 \times \dfrac {c + d} 2$

which may be a fair approximation if the field is approximately rectangular.

But the area of a quadrilateral depends not only on the lengths of its sides but also its angles.

The maximum area is obtained when the quadrilateral is cyclic is the area, in which case Brahmagupta's Formula can be used:

$\AA^2 = \paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d}$

which in this case gives approximately $1022$ square units.

So in this case the Egyptian Formula for Area of Quadrilateral works fairly well.

Sources

• 1992: John Hadley/2 and David Singmaster: Problems to Sharpen the Young (Math. Gazette Vol. 76, no. 475: pp. 102 – 126)  www.jstor.org/stable/3620384