# The Area of a Cyclic Quadrilateral

A quadrilateral inscribed is a circle is known as a cyclic quadrilateral. The proof is beyond the scope of this tutorial and will be discussed in an advanced tutorial, so only the formula is given here for application.

If $a$, $b$, $c$ and $d$ are the sides of a cyclic quadrilateral and if $s = \frac{{a + b + c + d}}{2}$, then

The area of a cyclic quadrilateral $= \sqrt {(s – a)(s – b)(s – c)(s – d)}$

Example:

In a circular grassy plot, a quadrilateral shape with its corners touching the boundary of the plot is to be paved with bricks. Find the area of the quadrilateral when the sides of the quadrilateral are $36$ m, $77$ m, $75$ m and $40$ m.

Solution:

Given that the sides of the quadrilateral are $a = 36$m, $b = 77$m, $c = 75$m and $d = 40$m
$s = \frac{{a + b + c + d}}{2} = \frac{{36 + 77 + 75 + 40}}{2} = \frac{{228}}{2} = 114m$

The area of the cyclic quadrilateral $= \sqrt {(s – a)(s – b)(s – c)(s – d)}$
The area of the cyclic quadrilateral $= \sqrt {(114 – 36)(114 – 77)(114 – 75)(114 – 40)}$
The area of the cyclic quadrilateral $= \sqrt {78 \times 37 \times 39 \times 74} = 39 \times 37 \times 2 = 2886$ square meters.