# Guide to find the area of Cyclic Quadrilateral with Brahmagupta’s formula

This article will be multi-parted. We will discuss basic concepts and conclude with Brahmagupta’s formula to find the area of a cyclic quadrilateral.

So, let’s get started with the concept of Brahmagupta’s **Area of a Quadrilateral Calculator**.

**The flow of the article:**

- What is Area?
- What is Quadrilateral?
- What is Cyclic Quadrilateral
- Area of a Quadrilateral Calculator – We will see how Brahmagupta’s formula calculates the area of a cyclic quadrilateral.

**What is Area?**

In geometry, Area is the measure of space covered by a flat surface. The area is the quantity that expresses the amount of space contained within a set of boundaries of a 2-dimensional region, measured in square units, i.e. square cm, square feet, square inches, etc.

**What is Quadrilateral?**

A quadrilateral is a polygon with four sides. To understand quadrilaterals, we must know about polygons.

The term polygon is a union of two words POLY(many) GON(sides).

Any closed figure made up of at least three line segments and connected end-to-end is termed a polygon. Polygons are available in many shapes enclosed by different no of line segments. QUADRILATERAL is one of them.

The term “Quadrilateral” is a merger of two Latin words, Quadra(four) Latus(sides).

A Quadrilateral is a polygon, having four sides, four vertices, & 4 angles.

Below is an example of a quadrilateral for better understanding:

The quadrilateral, as mentioned above, has four sides AB, BC, CD, DA, and four angles and four vertices A, B, C, D.

The quadrilateral should be named ABCD, BCDA, CDAB, OR DABC. We cannot call it ACBD OR CABD since it disturbs the order of vertices.

**What is a cyclic quadrilateral?**

It is a 4-sided polygon whose vertices lie on the circumference of a circle. The below-given figure would help in better understanding of Cyclic Quadrilateral.

Here, ABCD is a cyclic quadrilateral with a, b, c, & d as its side-lengths and p & q are its diagonals.

Some properties of cyclic quadrilaterals:

- All the four vertices of the quadrilateral lie on the circumference of the circle in a cyclic quadrilateral.
- The measure of an exterior angle at a vertex is equal to the opposite interior angle.
- The total value of all the angles in a cyclic quadrilateral is equal to 360°.
- The Sum of pairs of opposite angles is 180°; in the above-given figure, ∠A and ∠C = 180° and ∠D and ∠B = 180°.
- In a cyclic quadrilateral, the product of diagonals is equal to the sum of the product of opposite sides, which can be expressed as p × q = (a × c) plus (b × d).
- The four inscribed sides of the circle are the four chords of the circle. The chord of the circle is a line segment that joins two points of the circumference of the circle.

Area of quadrilateral calculator- Application of Brahmagupta’s formula in calculating the area of a cyclic quadrilateral.

Before divulging upon Brahmagupta’s formula, let’s get a brief about Brahmagupta.

Brahmagupta (598 C.E to 668 C.E) was an ancient Mathematician and astronomer.

Among many of his works on various topics, we can say that his formula of calculating the area of a cyclic quadrilateral is probably the most famous.

Here is Brahmagupta’s formula of calculating the area of a cyclic quadrilateral :

- Area = K = (s-a)(s-b)(s-c)(s-d)

S= a + b + c + d2

2. Perimeter = 2S = a + b +c +d

According to this formula, the area of Cyclic quadrilateral ABCD is the square root of the product of halves of the sums of the sides diminished by each side of the quadrilateral.

We have to find the total value of all sides and divide it in half. Then do the square root of the product of halves diminished by each side of the quadrilateral, as shown in formula 1.

Perimeter would be double of S OR the total of all sides.

Below are one example of the application of this formula and the properties of a cyclic quadrilateral:

**Example 1. In a circular playing ground, a quadrilateral with its corners touching the ground boundary is fenced with a quill. Find the area of the quadrilateral when the sides of the quadrilateral are 34 m, 75 m, 73 m, and 38 m.**

Solution: We have the information that the sides of the quadrilateral are a=34m, b=75m, c=73m, and d=38m

So, S = a + b + c + d2 = 34 + 75 + 73 + 382= 2202= 110m

Area of a quadrilateral = (s-a)(s-b)(s-c)(s-d)

Area of the cyclic quadrilateral = (110-34)(110-75)(110-73)(110-38)

Area of the cyclic quadrilateral =

76 353772 = 2662 square meters.

The perimeter would be double S which is 2(110)= 220m.

Hence, the area and perimeter of the cyclic quadrilateral are 2662 square meters and 228m, respectively.

**Example 2. Find the value of angle D of a cyclic quadrilateral if angle B is 80o.**

Solution: If ABCD is a cyclic quadrilateral, the sum of a pair of two opposite angles will be 180°.

∠B + ∠ D = 180°

80° + ∠D = 180°

∠D = 180° – 80°

∠D = 100°

The value of angle D is 120°.

**Example 3: Find the perimeter of a cyclic quadrilateral with sides 2cm, 6 cm, 8 cm, and 10 cm. **

Solution: Given the measurement of the sides are,

2cm, 6 cm, 8 cm and 10 cm

Using the formula of the perimeter,

Perimeter = 2s

S = a + b + c + d2

S = 2 + 6 + 8 + 102= 13.

Therefore, the perimeter of this cyclic quadrilateral would be 26cm.

**Example 4. If ABCD is a cyclic quadrilateral, ∠C=80∘, Then find the rest of the angles.**

Solution. If ∠C=80∘, Then ∠A=80∘, Since opposite angles in a cyclic quadrilateral are equal.

Now, as we know ∠B=∠D, and the total sum of all interior angles is 360°.

Hence, we can say that ∠A+∠B+∠C+∠D=360°

160°+2∠D = 360°

∠D=100°

Therefore, ∠A=80° ∠B=100° ∠C=80° ∠D=100°.

**Conclusion**

After reading this article, anybody would explain the area, quadrilateral, and cyclic quadrilateral. Also, one can understand Brahmagupta’s formula for calculating the area of a cyclic quadrilateral. The examples given above would further clarify the understanding of the formula and properties of a cyclic quadrilateral.