A quadrilateral is a shape in geometry that has four sides, four corners (called vertices), four angles, and two diagonals. One special type of quadrilateral is the cyclic quadrilateral. In a cyclic quadrilateral, all four corners lie exactly on the edge of a circle. This means you can draw a circle that passes through all four points of the shape. That circle is called the circumcircle of the quadrilateral. Because the vertices are on the circle, the shape is called “cyclic” (related to a circle). Not all quadrilaterals are cyclic, but if one is, it has some special properties, like the fact that the opposite angles always add up to 180°. These properties make cyclic quadrilaterals useful and important in geometry. So, in simple terms, a cyclic quadrilateral is a four-sided shape whose corners all touch the same circle.

In this Maths article, we will look into the Cyclic Quadrilateral introduction, properties, area, theorems, and solved examples in detail.

What is Cyclic Quadrilateral?

A cyclic quadrilateral is a four-sided shape where all the corners (or vertices) lie on a circle. This means that you can draw a single circle that passes through all four points of the shape. That circle is called a circumcircle, and the quadrilateral is said to be inscribed in the circle. The sides of the quadrilateral act like chords of the circle, connecting the points on the edge. This type of quadrilateral has special properties, such as the sum of its opposite angles always being 180 degrees.

The term “inscribed quadrilateral” is also used to describe the cyclic quadrilateral.

Properties of Cyclic Quadrilateral

These are the properties of a cyclic quadrilateral.

• A cyclic quadrilateral always has vertices that are on the circle’s circumference.
• The exterior angle created by extending any one side is equal to the interior angle on that side’s opposite side added together.
• The product of the diagonals in a cyclic quadrilateral is equal to the sum of the products of its two pairs of opposite sides.
• If only one side is created in a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
• The perpendicular bisectors of a cyclic quadrilateral are always concurrent.
• A rectangle or a parallelogram is created when the midpoints of the four sides of a cyclic quadrilateral are joined.

Area of a Cyclic Quadrilateral

The area of a cyclic quadrilateral is:

\(\sqrt{\left ( s-a \right )\left ( s-b \right )\left ( s-c \right )\left ( s-d \right )}\)

where

s= semi-perimeter

a, b, c, d are the quadrilateral’s four sides

Learn about Secant of a Circle and Area of a Quadrilateral.

Cyclic Quadrilateral: Angles, Radius, Diagonals, and Area

A cyclic quadrilateral is a four-sided figure where all the vertices lie on a single circle.

Angles in a Cyclic Quadrilateral

In a cyclic quadrilateral, the sum of the opposite angles is always 180°.

Let’s say the angles are ∠A, ∠B, ∠C, and ∠D, then:

• ∠A + ∠C = 180°
• ∠B + ∠D = 180°

Also, the total of all four angles is:

• ∠A + ∠B + ∠C + ∠D = 360°,
  which is the normal angle sum for any quadrilateral.

Radius of the Circle

If the sides of a cyclic quadrilateral are a, b, c, and d, and s is the semi-perimeter (half of the sum of all sides), then the radius R of the circle that passes through all four corners can be found using a formula (not shown here for simplicity).
The semi-perimeter is calculated as:

• s = ½ (a + b + c + d)

Diagonals of a Cyclic Quadrilateral

If a, b, c, and d are the sides and p and q are the lengths of the diagonals, then there are special formulas to calculate the lengths of the diagonals based on the sides. These formulas help when you don’t know the diagonals but know all sides.

Area of a Cyclic Quadrilateral

To find the area of a cyclic quadrilateral when you know all four sides:

Use Brahmagupta’s Formula:

• Area = √[(s - a)(s - b)(s - c)(s - d)]

Here, s is the semi-perimeter:

• s = ½ (a + b + c + d)

This formula works only if the quadrilateral is cyclic (its corners lie on a circle).

Theorems on Cyclic Quadrilateral

Several theorems regarding the cyclic quadrilateral are given below

Theorem: 1

A cyclic quadrilateral’s angle sum of either pair of opposite angles is supplementary.

Consider the cyclic quadrilateral ABCD enclosed in an O-centered circle.

Now connect O to the vertices A and C.

Consider the arc. ABC

\(\angle AOC = 2 \angle ABC = 2a\)

The angle that the same arc subtends is half of the angle subtended at its center.

now, consider the arc ADC

Reflex \(\angle AOC = 2 \angle ADC = 2\beta\)

Therefore,

\(\angle AOC + Reflex\, \angle AOC = 360^{\circ}\)

\(2\angle ABC + 2\angle ADC = 360^{\circ}\)

= \(2a + 2\beta = 360^{\circ}\)

= \(a + \beta = 180^{\circ}\)

Similar to this, a quadrilateral is cyclic if its opposing angles are supplementary.

Theorem: 2

The sum of the products of the two pairs of opposite sides makes up the product of the diagonals in a cyclic quadrilateral.

Let PQRS be a cyclic quadrilateral with PQ and RS, QR and PS, and PR and QS, respectively, as the opposite sides and diagonals.

so the product of diagonals is

\(PR \times QS = \left [ PQ \times RS \right ] + \left [ QR \times PS \right ]\)

Diagonal Ratio is provided by,

\(\frac{PR}{QS} = \left [ RS \times RQ \right ] + \frac{\left [ PS\times PQ \right ]}{\left [ RS\times PS \right ]} +\left [ QR \times PQ \right ]\)

Learn about Difference Between Circle and Sphere

Ptolemy Theorem of Cyclic Quadrilateral

When a quadrilateral is enclosed within a circle, the sum of the products of its two pairs of opposite sides equals the product of the diagonals.

AB and CD, as well as AD and BC, are the opposing sides of an ABCD quadrilateral if it is cyclic. Diagonals are AC and BD.

\(\left ( AB \times CD \right ) + \left ( AD\times BC \right ) = AC \times BD\)

Brahmagupta Theorem of Cyclic Quadrilateral

A cyclic quadrilateral’s area and perimeter can be calculated using this theorem. A cyclic quadrilateral with sides a, b, c, and d is represented by the area “K” as follows:

K = While 2S is used to represent the quadrilateral’s perimeter, S is the semi-perimeter. It comes from,

\(S = \left [ \frac{1}{2} \right ]\times \left [ a+b+c+d \right ]\)

Summary of Cyclic Quadrilateral

• A quadrilateral is cyclic if all four of its vertices are located on the circumference of a circle.
• Like all other quadrilaterals, a cyclic quadrilateral also has an angle sum of 360 degrees.
• Opposite pairs of angles are supplementary to each other” is the main characteristic of cyclic quadrilaterals.
• In a cyclic quadrilateral, the product of two opposite angles is 180 degrees.

Important Points about Cyclic Quadrilateral

• In a cyclic quadrilateral (a four-sided figure with all corners on a circle), the sum of opposite angles is always 180°. This means:
  • ∠A + ∠C = 180°
  • ∠B + ∠D = 180°
• Let’s say the quadrilateral has vertices A, B, C, and D.
  • The sides are:
    a = AB,
    b = BC,
    c = CD,
    d = DA
  • The diagonals are:
    p = AC and
    q = BD
• There's a formula that relates the diagonals to the sides:
  p × q = (a × c) + (b × d)
  This helps when you're trying to find the length of a diagonal using the sides.

Solved Examples of Cyclic Quadrilateral

Example 1. A circular grassy area is bordered by a quadrilateral cricket pitch with sides measuring 23, 54, 13, and 51 meters, respectively. How do you figure out the size of this pitch’s quadrilateral shape?

Solution: Its given

a = 23 m, b = 54 m, c = 13 m, d = 51 m.

s = \(\frac{\left ( a+b+c+d \right )}{2}\)

s = \(\frac{\left ( 23+54+13+51 \right )}{2}\)

s = 70.5 m

Since

\(k = \sqrt{\left ( s-a \right )\left ( s-b \right )\left ( s-c \right )\left ( s-d \right )}\)

\(k = \sqrt{\left ( 70.5- 23\right )\left ( 70.5- 54 \right )\left ( 70.5 – 13 \right )\left ( 70.5- 51 \right )}\)

\(k = \sqrt{47.5\times 16.5\times 57.5 \times 19.5}\)

\(k = \sqrt{878779.68}\)

\(k = 937.43 m^{2}\)

Example 2: In the below figure, determine the value of \(x^{\circ} \) and \(y^{\circ}\)

Solution: When one side of a cyclic quadrilateral is created, the exterior angle equals the interior opposite angle thanks to the exterior angle property of a cyclic quadrilateral.

So, we get

\(y^{\circ} = 100^{\circ}\)

\(x^{\circ} + 30^{\circ} = 60^{\circ}\)

\(x^{\circ} = 60^{\circ} – 30^{\circ}\)

As a result

\(x^{\circ} = 30^{\circ}\)

\(y^{\circ} = 100^{\circ}\)

Example 3: If angle B is 80 degrees, what is the value of angle D in the cyclic quadrilateral ABCD?

Solution: The sum of a pair of two opposite angles for a cyclic quadrilateral is 180 degrees, so

The ABCD cyclic quadrilateral

\(\angle B+ \angle D = 180^{\circ}\)

\(80^{\circ} + \angle D = 180^{\circ}\)

\(\angle D = 180^{\circ} – 80^{\circ}\)

\(\angle D = 100^{\circ}\)

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