# How to find the area of a quadrilateral given only the length of it's 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 it's 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?

• – mathlove Sep 2 '14 at 11:59

## 2 Answers

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)

• See also math.stackexchange.com/a/832795/589. – lhf Sep 2 '14 at 11:55
• I see. But if a diagonal is given then it'll be rigid with a unique area right? – Nabigh Sep 2 '14 at 12:04
• @Nabigh, yes: if a diagonal is given, then you'll have two triangles, and triangles are rigid. – lhf Sep 2 '14 at 12:05
• Thanks @lhf, that settles the problem. – Nabigh Sep 2 '14 at 12:16

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.