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


cyclic quadrilateral

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 = 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

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.