A quadrilateral whose vertices lie on a single circle is called cyclic quadrilateral. This circle is called the circumcircle, and the vertices are known to be concyclic.

The sum of the products of opposite sides of a cyclic quadrilateral is equal to the product of the two diagonals. The opposite angle of a cyclic quadrilateral is supplementary.

The formula for the area of a cyclic quadrilateral is:

âˆš(sâˆ’a) (sâˆ’b) (sâˆ’c) (sâˆ’d)

Where “s” is called the semi-perimeter,

s = a + b +c + d / 2

Example 1

Calculate the area of the quadrilateral when the sides of the quadrilateral are 30 m, 60 m, 70 m and 45 m.

Solution

Given parameters are

a=30m

b = 60m

c = 70m

d = 45m

s = a + b +c + d / 2

s = 30+60+70+45 / 2

s = 102.5 m

Area of cyclic quadrilateral is given by

= âˆš (sâˆ’a) (sâˆ’b) (sâˆ’c) (sâˆ’d)

= âˆš(102.5âˆ’30)(102.5âˆ’60)(102.5âˆ’70)(102.5âˆ’45)

= 2399.60 square meter

Example 2

Calculate the area of the quadrilateral when the sides of the quadrilateral are 35 m, 77 m, 75 m and 43 m.

Solution

Given parameters are

a = 35 m

b = 77 m

c = 75 m

d = 43 m

s = a + b + c + d / 2

s = 35 + 77 + 75 + 43 / 2

s = 115 m

Area of cyclic quadrilateral is given by

âˆš(sâˆ’a) (sâˆ’b) (sâˆ’c) (sâˆ’d)

= âˆš (115âˆ’35)(115âˆ’77)(115âˆ’75)(115âˆ’43)

2958.91 square meter