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Area of Cyclic Quadrilateral Formula

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  • Last Updated : 02 May, 2022

All four vertices of a cyclic quadrilateral lie on the circumference of a circle. Circumcircle is another name for a cyclic quadrilateral. A cyclic quadrilateral is defined as a quadrilateral with all four vertices on the circumference of a circle. The concept of cyclic quadrilateral will be discussed in-depth in this article.

Cyclic Quadrilateral

The word ‘quadrilateral’ is made up of two Latin words ‘quadri’ which means ‘four,’ and latus, which means ‘side’. As a result, a quadrilateral is a four-sided polygon with four edges and four vertices. A cyclic quadrilateral, on the other hand, is a specific sort of quadrilateral that has all four vertices on the circumference of a circle.

The circumcircle is noted for having concyclic vertices. The product of the two diagonals is equal to the total of the products of opposite sides of a cyclic quadrilateral. A cyclic quadrilateral’s supplemental angle is the opposite angle. A quadrilateral enclosed in a circle is known as a cyclic quadrilateral, sometimes known as a circumcircle or circumscribed circle. 

A cyclic quadrilateral is defined as a quadrilateral with all four vertices on the circumference of a circle.

Cyclic Quadrilateral Formula

A cyclic quadrilateral’s area is calculated using the following formula:

k = √(s-a)(s-b)(s-c)(s-d)

Where,

  • s = Semi perimeter [s = (a + b + c + d) / 2]
  • a, b, c, d = Sides of Cyclic Quadrilateral

Sample Questions

Question 1: Is it possible to regard a parallelogram to be a cyclic quadrilateral?

Answer:

When the opposite angles of a parallelogram are supplementary and all four vertices of the same parallelogram lie on the circumference of a circle, the parallelogram is called a cyclic quadrilateral.

Question 2: In a cyclic quadrilateral, what are the qualities of perpendicular bisectors?

Answer:

The qualities of perpendicular bisectors of cyclic quadrilateral:

  • A cyclic quadrilateral’s perpendicular bisectors are always at the same place.
  • In a cyclic quadrilateral, the perpendicular bisectors of all four sides meet in the circle’s center.

Question 3: Calculate the area of a cyclic quadrilateral with sides of 21 meters, 35 meters, 62 meters, and 12 meters.

Answer:

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

Question 4: 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?

Answer:

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

Question 5: The sides of a cyclic quadrilateral are 28 m, 61 m, 37 m, 10 m then calculate its area.

Answer:

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

Question 6: How do you calculate the perimeter of a cyclic quadrilateral with sides of 12 cm, 21 cm, 10 cm, and 5 cm?

Answer:

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

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