# How to find the area of a quadrilateral given only the length of its sides?

How do you find the area of a convex quadrilateral $$ABCD$$, given only the length of its sides $$a$$, $$b$$, $$c$$ and $$d$$. If the length of a single diagonal is given, I could easily find its area by dividing it into two triangles and applying Heron's formula to each.

I encountered this problem while trying to find the area of a patch of land. Sometimes, the area maybe approximated by a rectangle or a trapezium or some other simple figure, but in general opposite sides are not parallel. All the formulas I have so far seen includes knowing at least one angle or a diagonal.

So is there a formula (even a complicated one maybe) that gives the area of a quadrilateral given only its side lengths?

• Sep 2, 2014 at 11:59

Quadrilaterals are not rigid: you can fix a side and move one of the other vertices freely while respecting all side lengths. The area changes when you move the vertex.

(image from http://www.mathsisfun.com/definitions/rigid.html)

Here you see that the area is not determined. The animation uses parallelograms with side lengths $a=c=3$, $b=d=2$, so the area varies between $0$ and $6$. In general the minimum area may not be exactly zero as here, but the same thing happens always.