How to find the area of a quadrilateral with all sides different?

Question

How to find the area of a quadrilateral in which the length of all the sides are different. For example one pair of opposite sides are 630 foot and 357 foot, and another pair of opposite sides are 587 foot and 358 foot. Furthermore, no measure of any angle is given. The figure is roughly drawn as follows.

Answer 1

There are two particular cases where the area of a quadrilateral is determined solely by its sides $a,b,c,d$:

• when it possesses a circumscribed circle. In this case, the quadrilateral is called cyclic and we have to use Brahmagupta formula:

$$A={\frac{1}{4}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}}.$$

giving the following area...

$$A \approx 217,144 ft^2$$

This area is achievable (see quadrilateral ABCD on fig. 1)

• when the quadrilateral possesses an inscribed circle ; in this case, it is called a tangential quadrilateral.

The formula for the area of a tangential quadrilateral is:

$$\displaystyle A=\sqrt{(e+f+g+h)(efg+fgh+ghe+hef)}.$$

with $$\begin{cases}e&+&f&&&&&=&a\\ &&f&+&g&&&=&b\\ &&&&g&+&h&=&c\\ e&+&&&&&h&=&d\end{cases}$$

giving the condition $a+c=b+d$ which isn't fullfilled (the sums of opposite sidelengths aren't equal)

Remark: the area quadrilateral ABCD can have is $\approx 218,700$ (obtained experimentaly).

In order to understand how, with the same lengths, the area of quadrilateral $ABCD$ can vary, have a look at the following figure, where AB (length 630) is fixed. .

Fig. 1. For a fixed line segment $AB$, vertices $C$ and $D$ belong to circular arcs with $CD=357$. A particular case occurs when $B,C,D$ are aligned. Two cases are highlighted : the case where $ABCD$ is cyclic and a non convex case.

Answer 2

Quadrilaterals are, in general, not determined by solely their side lengths. If you are given a quadrilateral and you only know the side lengths, there are infinite possible areas it can have.

The only case where you could determine the area is if the quadrilateral was degenerate. That is, if the sides are $a,b,c,d$ with WLOG $\text{max}(a,b,c,d)=a$, then we have iff $a=b+c+d$, the area of the quadrilateral is $0$.

The question can be solved, however, if there is more information provided. For example, an angle measure or diagonal length and it's location wrt the sides.

Answer 3

If you imagine your quadrilateral as built from rods of the prescribed lengths joined at the corners in a way that does not constrain the angles and fix one side as a "hoirizontal base" you can see how the other two vertices can move to change the area.

There are pictures in the answer from @JeanMarie.