Area of 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 discuss in advanced tutorial, only the formula is given for application.
If a, b, c and dare the sides of a cyclic quadrilateral and if s = \frac{{a + b + c + d}}{2}, then


cyclic quadrilateral

Area of 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 36m, 77m, 75m and 40m.

Solution:

Given the sides of the quadrilateral are a = 36m, b = 77m, c = 75m and d = 40m

s = \frac{{a + b + c + d}}{2} = \frac{{36 + 77 +  75 + 40}}{2} = \frac{{228}}{2} = 114m


Area of cyclic quadrilateral  = \sqrt {(s - a)(s - b)(s - c)(s - d)}
Area of cyclic quadrilateral  = \sqrt {(114 - 36)(114 - 77)(114 - 75)(114 - 40)}
Area of cyclic quadrilateral  = \sqrt {78 \times 37 \times 39 \times 74}  = 39 \times 37 \times 2 = 2886 Square meter.