Table of Content

• Cyclic Quadrilateral Definition
• Angles in Cyclic Quadrilateral
• Properties of Cyclic Quadrilateral
• Area of Cyclic Quadrilateral Formula
• Theorem on Cyclic Quadrilateral

Property	Description
Sum of Opposite Angles	The sum of the measures of opposite angles is always 180 degrees.
Sum of Adjacent Angles	The sum of adjacent angles in a cyclic quadrilateral is always 180 degrees.
Exterior Angle Sum	The exterior angle formed by extending one side of the cyclic quadrilateral equals the interior opposite angle.
Diagonals	The product of the lengths of the diagonals equals the sum of the products of opposite sides.
Opposite Angles	Opposite angles of any cyclic quadrilateral are supplementary.
Equal Opposite Angles Bisectors	If two opposite angles in a cyclic quadrilateral are bisected by the diagonal, then the point of intersection of the bisectors lies on the circumcircle.
Perpendicular Bisectors of Sides	The perpendicular bisectors of the sides of a cyclic quadrilateral intersect at a point that lies on the circumcircle of the quadrilateral.
Sum of Squares of Sides	The sum of the squares of the sides equals the sum of the squares of the diagonals plus four times the square of the radius of the circumcircle.

Cyclic Quadrilateral Formula

There are various formulas given for Cyclic Quadrilateral, some of the important ones are:

• Area of Cyclic Quadrilateral
• Radius of Circumcircle
• Diagonals of Cyclic Quadrilateral

Let’s discuss these formula in detail as follows:

Area of Cyclic Quadrilateral Formula

Area of the cyclic quadrilateral is calculated using the following formula:

Area of Cyclic Quadrilateral = √(s-a)(s-b)(s-c)(s-d)

Where,

• a, b, c, d are the sides of Cyclic Quadrilateral, and
• s is semi perimeter [s = (a + b + c + d) / 2].

Note: This formula is also known as Brahmagupta’s Formula.

Radius of Circumcircle

Let the sides of  a cyclic quadrilateral be a, b, c and d, and s is the semi perimeter, then the radius of circumcircle is given by,

R = 1/4 [Tex]\bold{\sqrt{\frac{(ab + cd)(ac + bd)(ad + bc)}{(s-a)(s-b)(s-c)(s-d)}}}[/Tex]

Diagonals of Cyclic Quadrilaterals

Diagonal is the line in any polygon which joins any two non-adjacent vertices.

Suppose a, b, c and d are the sides of a cyclic quadrilateral and p & q are the diagonals, then we can find the diagonals of it using the below-given formulas:

p = [Tex]\bold{\sqrt{\frac{(ac + bd)(ad + bc)}{ab + cd}}}[/Tex] and q = [Tex]\bold{\sqrt{\frac{(ac + bd)(ab + cd)}{ad + bc}}}[/Tex]

Theorem on Cyclic Quadrilateral

To understand the Cyclic Quadrilateral better, we look at different theorems in geometry. Some of these important theorems are:

• Inscribed Angle Theorem
• Ptolemy’s Theorem

Now, let’s delve into these theorems in detail:

Inscribed Angle Theorem

According to Inscribed Angle Theorem,

Sum of the opposite angle of any cyclic quadrilateral is supplementary.

Given: A cyclic quadrilateral ABCD inside a circle with centre O.

Construction: Join the radius OA and OC

Proof:  In quadrilateral ABCD,

2 × ∠ABC = Reflex ∠ AOC  (According to Circle Theorem)…(eq. 1)

Similarly,

 2 × ∠ADC = ∠ AOC…(eq. 2)

We know that,

∠ AOC + Reflex ∠ AOC = 360°…(eq. 3)

By eq (1) + eq (2)

2 × ∠ADC + 2 × ∠ABC = Reflex ∠ AOC + ∠ AOC

2 × (∠ADC + ∠ABC) = Reflex ∠ AOC + ∠ AOC

2 × ∠ADC + ∠ABC = 360°    (by eq. 3)

∠ADC + ∠ABC = 180° (supplementary)

Similarly,

∠BAD + ∠BCD = 180° (supplementary)

Thus, the opposite angles of a cyclic quadrilateral are supplementary.

Converse of the above Theorem is also true.

i.e., If the sum of opposite angles in a quadrilateral is supplementary, then it is a cyclic quadrilateral.

Ptolemy’s Theorem

Ptolemy’s Theorem is named after the Greek astronomer and mathematician Claudius Ptolemy (c. 100 – c. 170 AD). The theorem states:

For any cyclic quadrilateral, the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals.

Mathematically, if ABCD is a cyclic quadrilateral with sides AB, BC, CD, and DA, and diagonalsAC and BD, then Ptolemy’s Theorem can be expressed as:

AB ⋅ CD + BC ⋅ DA = AC ⋅ BD

OR

If a, b, c and d are the lengths of the sides of a cyclic quadrilateral, and e and f are the lengths of the diagonals, then:

ac + bd = ef

Conclusion

In conclusion, cyclic quadrilaterals are quadrilaterals with four vertices on one circle. They have special properties like perpendicular lines inside and Ptolemy’s theorem. They’re easy to make and help solve problems in geometry. Learning about them helps us understand shapes better and see how math is beautiful, encouraging us to learn more.

Read More,

• Quadrilateral
• Area of Quadrilateral
• Area of Triangle

Solved Examples of Cyclic Quadrilateral

Example 1: Calculate the area of a cyclic quadrilateral with sides of 21 meters, 35 meters, 62 meters, and 12 meters.

Solution:

Given: a = 21 m, b = 35 m, c = 62 m, d = 12 m.

s = (a + b + c + d) / 2

∴ s = (21 + 35 + 62 + 12) / 2

∴ s = 65 m

Since,

k = √(s – a)(s – b)(s – c)(s – d)

∴ k = √(65 – 21)(65 – 35)(65 – 62)(65 – 12)

∴ k = √44 × 30 × 3 × 53

∴ k = √209880

∴ k = 458.12 m²

Example 2: A quadrilateral cricket pitch with sides of 23 m, 54 m, 13 m, and 51 m touches the limits of a circular grassy area. How do you calculate the area of this quadrilateral-shaped pitch?

Solution:

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

s = (a + b + c + d) / 2

∴ s = (23 + 54 + 13 + 51) / 2

∴ s = 70.5 m

Since,

k = √(s – a)(s – b)(s – c)(s – d)

∴ k = √(70.5 – 23)(70.5 – 54)(70.5 – 13)(70.5 – 51)

∴ k = √47.5 × 16.5 × 57.5 × 19.5

∴ k = √878779.68

∴ k = 937.43 m²

Example 3: The sides of a cyclic quadrilateral are 28 m, 61 m, 37 m, and 10 m then calculate its area.

Solution:

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

s = (a + b + c + d) / 2

∴ s = (28 + 61 + 37 + 10) / 2

∴ s = 68 m

Since,

k = √(s – a)(s – b)(s – c)(s – d)

∴ k = √(68 – 28)(68 – 61)(68 – 37)(68 – 10)

∴ k = √40 × 7 × 31 × 58

∴ k = √503440

∴ k = 709.53 m²

Example 4: How do you calculate the perimeter of a cyclic quadrilateral with sides of 12 cm, 21 cm, 10 cm, and 5 cm?

Solution:

Given: a = 12 cm, b = 21 cm, c = 10 cm, d = 5 cm.

s = (a + b + c + d) / 2

∴ s = (12 + 21 + 10 + 5) / 2

∴ s = 24 cm

perimeter of a cyclic quadrilateral = 2s

∴ perimeter of a cyclic quadrilateral = 2 × 24

∴ perimeter of a cyclic quadrilateral = 48 cm

Example 5: Find the value of ∠A in a cyclic quadrilateral, if ∠C is 70°.

Solution:

For cyclic quadrilateral ABCD, the sum of the pairs of two opposite angles is 180°.

∠A + ∠C = 180°  

70° + ∠C = 180°

∠C = 180° – 70°

∠C = 110°

The value of the angle C is 120°.

Example 6: ABCD is a Cyclic Quadrilateral with sides a, b, c and d & diagonal p and q, then how to calculate the length of diagonals?

Solution:

For cyclic quadrilateral ABCD with side a, b, c and d, length of diagonal is given by:

p = [Tex]\bold{\sqrt{\frac{(ac + bd)(ad + bc)}{ab + cd}}}[/Tex] and q = [Tex]\bold{\sqrt{\frac{(ac + bd)(ab + cd)}{ad + bc}}}[/Tex]

Using these formulas we can easily calculate the length of diagonals.

FAQs on Cyclic Quadrilateral

What is Cyclic Quadrilateral?

The quadrilateral whose all four vertices lie on the circumference of a circle is called a cyclic quadrilateral. Another name for it is the inscribed quadrilateral.

What is Brahmagupta’s Formula?

For cyclic quadrilateral with sides a, b, c & d, Brahmagupta’s Formula for calculating area is given by

Area of Cyclic Quadrilateral = √(s-a)(s-b)(s-c)(s-d)

Where s is semiperimeter.

Sum of the Opposite Angles of the Cyclic Quadrilateral is equals to _____.

Sum of the opposite angles of the cyclic quadrilateral is equals to 180°.

How to Calculate Area of Cyclic Quadrilateral?

To calculate the area of cyclic quadrilater we can use Brahmagupta’s Formula, which uses the length of sides and semiperimeter to calculate the area.

Is Parallelogram a Cyclic Quadrilateral?

Not, every parallelogram is a cyclic quadrilateral, for a parallelogram to be cyclic the opposite angles of that parallelogram should be supplementary.

What is Sum of Oppotise Angles in Cyclic Quadrilateral?

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