What is Cyclic Quadrilateral

A cyclic quadrilateral is a special type of quadrilateral in which all four vertices lie on the circumference of a circle. In other words, a quadrilateral is called cyclic if a single circle can be drawn that passes through all its vertices. Such a quadrilateral is said to be inscribed in a circle, and its vertices are called concyclic points. The center of the circle is known as the circumcenter, and the radius of the circle is called the circumradius.

In the figure given below, ABCD is a cyclic quadrilateral with a, b, c, and d as the side lengths.

Angles in Cyclic Quadrilateral

A sum of the opposite angles of a cyclic quadrilateral is supplementary.

For a cyclic quadrilateral, let the internal angles be ∠A, ∠B, ∠C, and ∠D Then,

• ∠A + ∠C = 180°... (1)
• ∠B + ∠D = 180°... (2)

Adding equation (1) and (2), we get

∠A + ∠B + ∠C + ∠D = 360°

Thus, the angle-angle sum property of a quadrilateral also holds true for a cyclic quadrilateral.

Properties of Cyclic Quadrilateral

A cyclic quadrilateral is a special quadrilateral in which all its vertices lie on the circumference of a circle. Some of the important properties of a cyclic quadrilateral are:

Cyclic Quadrilateral Formula

There are various formulas given for cyclic quadrilaterals; some of the important ones are

1. Area of Cyclic Quadrilateral Formula

The 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.

2. Radius of Circumcircle

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

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

3. 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 = \bold{\sqrt{\frac{(ac + bd)(ad + bc)}{ab + cd}}} and q = \bold{\sqrt{\frac{(ac + bd)(ab + cd)}{ad + bc}}}

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

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 center 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 diagonals AC and BD, then Ptolemy's Theorem can be expressed as:

AB ⋅ CD + BC ⋅ DA = AC ⋅ BD

OR

If a, b, c, 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

Also Check

• 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 & diagonals p and q. Then how do you calculate the length of the diagonals?

Solution:

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

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

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