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

== Historical Note on by : Problem $23$: De Campo Quadrangulo == The formula being used here for a quadrilateral whose sides are $a$, $b$, $c$ and $d$ is the Roman and Egyptian one:
 * $\AA = \dfrac {a + b} 2 \times \dfrac {c + d} 2$

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 Roman-Egyptian formula works fairly well.