Cyclic Quadrilateral

What is Cyclic Quadrilateral?

  • If the four vertices of a quadrilateral lie on the circumference of the circle then it is known as a cyclic quadrilateral.

  • A cyclic quadrilateral is also known as a circumcircle or a circumscribed cycle.

  • The vertices of a quadrilateral are said to be concyclic.

  • Examples of cyclic quadrilaterals.                                                                     

What is a Quadrilateral?

  • A quadrilateral is a polygon having four edges and four corners or vertices.

  • The sum of interior angles of a quadrilateral sum up to 360 degrees.

  • The word ‘quadrilateral’ is composed of two Latin words, Quadri which means ‘four’ and latus which means ‘side’.

  • A quadrilateral is a two-dimensional figure having four edges.

Properties of Cyclic Quadrilateral –

Here’s a table that lists down the properties of a cyclic quadrilateral.

  • The opposite angles of a cyclic quadrilateral are supplementary which means that the sum of either pair of opposite angles is equal to 180 degrees.

  • The four perpendicular bisectors in a cyclic quadrilateral meet at the centre.

  • A quadrilateral is said to be cyclic if the sum of two opposite angles are supplementary.

  • The perimeter of a cyclic quadrilateral is 2s, where s = semi perimeter 


  •                       \[s = \frac{{a + b + c + d}}{2}\]


    • The area of a cyclic quadrilateral is = \[ = \frac{1}{2}\left[ {s\left( {\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \right)} \right]\]

          where, a, b, c and d are the four sides of a quadrilateral.

    • A cyclic quadrilateral has four vertices which lie on the circumference of the circle.

  • If you just join the midpoints of the four sides in order in a cyclic quadrilateral, we get a rectangle or a parallelogram.

  • The perpendicular bisectors are concurrent in a cyclic quadrilateral.

  • If A, B, C and D are four sides of a quadrilateral and E is the point of intersection of the two diagonals in the cyclic quadrilateral then 

  • AE × EC = BE × ED


    Formula for the Perimeter of a Cyclic Quadrilateral -

    The perimeter of a cyclic quadrilateral is 2s, where s = semi perimeter 


                         \[s = \frac{{a + b + c + d}}{2}\]


    Perimeter can be simplified in the following way,

    Here, a+c = b+d,

              Substituting in the formula for semi perimeter we get, \[s = \frac{{b + d + b + d}}{2}\]

    \[s = \frac{{2\left( {b + d} \right)}}{2}\]

    s= b + d




    Formula for the Area of an Inscribed or Cyclic Quadrilateral -

    The area of a cyclic quadrilateral is = \[ = \frac{1}{2}\left[ {s\left( {\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \right)} \right]\]

          where, a, b, c and d are the four sides of a quadrilateral.


    Cyclic Quadrilateral Theorem-

    Here is the important cyclic quadrilateral theorem

    1. Theorem of Cyclic Quadrilateral (I) 

    The either pair of the opposite angles of a cyclic quadrilateral sum up to 180° .

    Given, ABCD is a cyclic quadrilateral of a cycle with the centre as O.

            Here, \[\angle BAD + \angle BCD = {180^o}\]

                            \[\angle ABC + \angle ADC = {180^o}\]

               Therefore, cyclic quadrilateral angles equal to 180 degrees.

    1. Theorem of Cyclic Quadrilateral (II)

    In a cyclic quadrilateral, if a quadrilateral is inscribed inside a cycle, the product of the diagonals of the cyclic quadrilateral is equal to the sum of the two pairs of opposite sides of the cyclic quadrilateral.                                                      

    In a cyclic quadrilateral ABCD, AC and BD are diagonals and AB, CD, AD and BC are opposite sides.

    Product of Diagonals -

    \[\left( {AC \times BD} \right){\text{ }} = {\text{ }}\left( {AB \times CD} \right){\text{ }} + \left( {AD \times BC} \right)\]

    Ratio of Diagonals -

    \[\frac{{AC}}{{BD}} = \frac{{\left( {AB \times AD} \right) + \left( {BC \times CD} \right)}}{{\left( {AB \times BC} \right) + \left( {AD \times CD} \right)}}\]

    Questions on Cyclic Quadrilateral Angles and Based on Cyclic Quadrilateral Theorem-

    Question 1: What will be the value of angle B of a cyclic quadrilateral if the value of angle D is equal to 60 degrees.

    Solution: Let’s list down the given information, \[\angle B\] = 60°

    As quadrilateral ABCD is cyclic, which means that the sum of a pair of two opposite angles in a cyclic quadrilateral will be equal to 180° according to the cyclic quadrilateral theorem.

    \[\angle B\]+\[ \angle D\] = 1800

      600+ \[\angle D\] = 1800

      \[\angle D\] = 1800 - 600

    Therefore, the value of\[\angle D\] = 1200

    Question 2: What will be the value of angle D of a cyclic quadrilateral if the value of angle B is equal to 70 degrees.

    Solution:  Let’s list down the given information, \[\angle D = {70^o}\]

    As quadrilateral ABCD is cyclic, which means that sum of a pair of two opposite angles in a cyclic quadrilateral will be equal to 180° according to the cyclic quadrilateral theorem.

      \[\angle B\] + \[\angle D\] = 1800

      \[\angle B\] + 700 = 1800

      \[\angle B\] = 1800 - 700

    Therefore, the value of\[\angle D = {110^o}\]

    Question 3: Find the perimeter of a cyclic quadrilateral with sides 4cm, 2 cm, 6 cm and 8 cm. 

    Solution:  Given the measurement of the sides are,

    4cm, 2 cm, 6 cm and 8 cm

     Using the formula of perimeter,

      Perimeter = 2s            

      \[S = \frac{{a + b + c + d}}{2}\]

      \[S = \frac{{4 + 6 + 2 + 8}}{2}\] 

    s= 10 

    Therefore, perimeter of a cyclic quadrilateral = 2s = 20