Answer to Question #201716 in Geometry for Hussain Darboe

Question

Find the area of an irregular quadrilateral with sides of 22.20m, 8.75m, 8.24m and 15.30m?

Answer 1

It is not possible to find the area of an irregular quadrilateral with only the length of the 4 sides are given because the area can vary with the change of the angles between that sides. But we can find the limits of its change. The main formula for the area of a quadrilateral is Bresneider formula:

$$S^2=(s-a)(s-b)(s-c)(s-d)-abcd*cos^2((α+γ)/2)$$

  And it depends on the angles of the quadrilateral. The area tends to 0 when two opposite angles tend to 0 and the others – to 180 . We can find the maximal possible area of an irregular quadrilateral with the given 4 sides assuming that the difference between the angles tends to 0 and they tend to be equal.. In this case we can use Brahmagupta formula (S - area of the quadrilateral):

$$S^2=(s-a)(s-b)(s-c)(s-d)$$

p is perimeter of the quadrilateral, s is half-perimeter,  

$s=p/2=(22.20m + 8.75m + 8.24m + 15.30m)/2=54.49/2=27.245 (m)$

$$S^2=(27.245-22.2)(27.245-8.75)(27.245-8.24)(27.245-15.3)=$$

$$=1.227*18.495*19*11.945=5150.37(m^2)$$

Square root of 5150.37 is 71.77(m²)

Hence the area of ​​the quadrangle can vary from 0 to 71.77

Answer: the maximal area of the quadrilateral is 71.77 m²