Area of Quadrilateral: The Area of a quadrilateral is the space inside the boundary of a quadrilateral or in other words, the space enclosed by the edges of a quadrilateral. A quadrilateral is a closed two-dimensional shape with four sides or edges, and also four corners or vertices. In mensuration, the shape of objects is classified based on the number of sides of the polygon.

The area of a quadrilateral can be found by dividing it into two triangles using a diagonal. When the length of the diagonal and the heights of the two triangles are given, the area of the quadrilateral is, A = (1/2) × Diagonal × (Sum of heights).

Quadrilaterals or any polygons can be classified into two categories, regular quadrilaterals/polygons i.e., all sides are of equal length, and irregular quadrilaterals i.e., all sides are not equal. This article explores the Area of Quadrilateral, using different methods, along with its formula, solved examples, and practice problems.

What is a Quadrilateral?

A quadrilateral is a polygon with four sides. A closed two-dimensional figure, formed by joining the four non-collinear points is called a quadrilateral. A quadrilateral has four sides, four angles, and four vertices. The sides of the quadrilateral may or may not be equal. Various types of quadrilaterals can be defined based on the properties of their angles, sides, and diagonals, some of which are as follows:

• Rectangle
• Square
• Rhombus
• Parallelogram
• Trapezium
• Kite

Properties of Quadrilateral

All quadrilaterals have some common properties that are as follows:

• A closed figure has four sides.
• The summation of the Interior angles of a quadrilateral is 360 degrees.
• The four sides can vary in length or maybe equal depending upon the type of quadrilateral.

What is the Area of Quadrilateral?

Area of a quadrilateral is the space enclosed by all the boundaries of a quadrilateral. Area of a quadrilateral is measured in square units such as m², in², cm², etc. Area of a regular quadrilateral is calculated by using different formulas. For calculating the area of irregular quadrilateral various formulas are used which are discussed below in this article.

Area of the quadrilateral ABCD = (1/2) × d × h₁+ (1/2) × d × h₂ = (1/2) × d ×( h₁+h₂ ).

Thus, the formula used to find the area of a quadrilateral is,

Area of Quadrilateral = (1/2) × Diagonal × (Sum of heights) = (1/2) × d ×( h₁ + h₂ )

Area of Quadrilateral with Vertices

If vertices of a quadrilateral are given then its area is calculated by the given formula. Suppose A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), and D(x₄, y₄) be the vertices of a quadrilateral ABCD.

Then its area is calculated by using two different methods which are discussed below:

Area of Quadrilateral Using Coordinates

Follow the directions of the arrow, and add the diagonal products, i.e., x₁y₂, x₂y₃, x₃y₄, and x₄y₁.

(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁)….(i)

Now, follow the dotted arrows and add the diagonal products, i.e., x₂y₁, x₃y₂, x₄y₃, and x₁y₄.

(x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)….(ii)

Now, subtract equation (ii) from (i) and multiply the result by 1/2.

(1/2) × [(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) – (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)]

Thus, the formula for the area of the quadrilateral when vertices are given:

Area of Quadrilateral Using Area of Triangle

For this method, we divide the given quadrilateral into two triangles and then find the area of each triangle separately. At last, both the area of triangles are added to find the final area of the quadrilateral.

Area of quadrilateral ABCD = Area of triangle ABD + Area of triangle BCD

Area of a triangle with vertices P(x₁, y₁), Q(x₂, y₂), and R(x₃, y₃) is given by

Area of Quadrilateral Using Bretschneider′s Formula

When two opposite angles and all the sides of a quadrilateral are given, we can calculate its area using Bretschneider’s Formula which is the extension of heron’s formula for quadrilaterals and is given as follows:

How to find the Area of Quadrilateral?

Area of a quadrilateral is found by using the steps discussed below:

Step 1: Mark the length of the diagonal and the length of the perpendicular to it from both vertices.

Step 2: Put these values in the given formula Area = (1/2) × d ×(h₁+h₂ ), where d is the length of the diagonal and h_1, h₂ are lengths of the perpendicular from diagonal to opposite vertices.

Step 3: Answer obtained from the above step is the required area and is measured in unit²

Area of Some Quadrilaterals

Some specific quadrilaterals are very common and are used in our daily life and their formula for areas are explained in the article given below:

Area of a Square

A square is a special case of a rectangle in which the four sides are equal, and all the sides are parallel to each other. In a square diagonal bisect perpendicularly to each other.

Read More: Area of Square

Area of a Rectangle

A rectangle is a closed figure having four sides in which opposite sides are equal and parallel to each other and the diagonals of the rectangles bisect at 90 degrees.

Read More: Area of Rectangle

Area of Rhombus 

A Rhombus is a special case of the square in which all the four sides and opposite angles are the same in measure and the opposite sides are parallel and the sum of the adjacent angles of a rhombus is equal to 180 degrees.

Where D1 and D2 are the length of diagonals of Rhombus.

Read More: Area of Rhombus

Area of Parallelogram

The quadrilateral in which opposite sides are equal and parallel to each other is known as a parallelogram. In this, diagonals bisect each other and the opposite angles are of equal measure in which the sum of two adjacent angles of a parallelogram is equal to 180 degrees.

Read More: Area of Parallelogram

Area of Trapezium

This quadrilateral is somewhat different from the others as there is only one pair of the opposite side of a trapezium parallel to each other and the adjacent sides are supplementary to each other and the diagonals bisect each other in the same ratio.

Read More: Area of Trapezium

Area of Trapezium

Area of Kite

Kite is a special quadrilateral in which each pair of consecutive sides is congruent, but the opposite sides are not congruent. In this, the largest diagonal of a kite bisects the smallest diagonal.

Also Read: Area of Kite

where, D1 = long diagonal of kite(CD), D2 = short diagonal of kite(AB)

Area of Quadrilateral Solved Examples

Example 1: Find the area of the quadrilateral ABCD when its vertices are (1, 2), (5, 6), (4, −6), and (−5, 2).

Solution:

Let A(1, 2), B(5, 6), C(4, -6), and D(-5, 2) be the vertices of a quadrilateral ABCD.

A(1, 2) = (x₁, y₁), B(5, 6) = (x₂, y₂), C(4, -6) = (x₃, y₃), D(-5, 2) = (x₄, y₄)

We know that,

Area of Quadrilateral = (1/2) × [(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) – (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)]

⇒ Area of Quadrilateral = (½). {[1(6) + 5(-6) + 4(2) + (-5)2] – {[5(2) + 4(6) + (-5)(-6) + 1(2)]}

⇒ Area of Quadrilateral = (½).[(6 – 30 + 8 – 10) – (10 + 24 + 30 + 2)]

⇒ Area of Quadrilateral  = (½) [-26 – 66]

⇒ Area of Quadrilateral = 92/2 (area is never negative)

⇒ Area of Quadrilateral = 46 unit²

Example 2: Find the area of the trapezium if height is 5 cm and AB and CD are given as 10 and 6 cm respectively.

Solution: 

Given, AB = 10cm, CD = 6cm, height = 5cm

According to the formulae,

Area of Trapezium = (1/2) h (AB+CD)

⇒ Area of Trapezium = 1/2 x 5 x (10 + 6)

⇒ Area of Trapezium = 40 cm²

Example 3: Find the area of a kite whose longest and shortest diagonals are 20cm and 10cm respectively.

Solution: 

Length of longest diagonal, D₁= 20 cm

Length of shortest diagonal, D₂= 10 cm

So, Area of kite =1/2 x D₁ x D₂

⇒ Area of kite = 1/2 x 20 x 10

⇒ Area of kite  = 100 cm²

Example 4: Calculate the area of a parallelogram, if the base and height are 10 m and 15 m respectively.

Solution: 

Given, base = 10 m and height = 15 m

Area of Parallelogram = Base x Height

⇒ Area of Parallelogram = 10 x 15

⇒ Area of Parallelogram = 150 m²

Example 5: Given the area of the rhombus is 120-meter square then find the length of one of the diagonals if the other diagonal is of length 12 m.

Solution: 

Since we know that,

Area of Rhombus = (1/2) x Diagonal₁ x Diagonal₂ 

Putting all the known values, we get

120 = (1/2) x Diagonal 1 x Diagonal 2

Diagonal 2 = 20 m

Practice Problems on Area of Quadrilateral

1. Find the area of a parallelogram with base 6 cm and height 8 cm.

2. A rectangle has a length of 10 meters and a width of 4 meters. Calculate its area.

3. Calculate the area of a trapezoid with bases of lengths 6 cm and 10 cm, and a height of 5 cm.

4. Given that the diagonals of a rhombus are 8 cm and 10 cm, find the area of the rhombus.

5. You are given the coordinates of the vertices of an irregular quadrilateral: A(0,0), B(4,3), C(6,7), D(2,5). Calculate its area.

Area of Quadrilateral Class 8

The area of a quadrilateral depends on its shape and the information available about it. For Class 8, you’re likely dealing with specific types of quadrilaterals like rectangles, squares, parallelograms, trapezoids (or trapeziums, depending on the regional terminology), and kites.

Also Check:

• Quadrilaterals Class 8 Notes
• Quadrilaterals Class 9 Notes

Conclusion of Area of Quadrilateral

The area of a quadrilateral is important for many practical tasks in geometry, construction, and design. Each type of quadrilateral, like rectangles, squares, trapezoids, and parallelograms, has its own method for calculating area. Understanding these methods helps in making accurate measurements and solving real-world problems.

Area of Quadrilateral – FAQs

What is the area of a quadrilateral?

Area of the quadrilateral is the region inside the boundary of a quadrilateral. It is the total space occupied by a quadrilateral in 2-D plane. It is measured in square units.

How to find the area of a quadrilateral?

Area of quadrilateral is found using formula given below:

Area of Quadrilateral = (1/2) × [(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) – (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)]

where (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄) are the vertices of a quadrilateral.

What are the different types of quadrilaterals?

Different types of quadrilateral are:

• Square
• Rectangle
• Rhombus
• Kite
• Parallelogram
• Trapezium

Write the uses of quadrilaterals.

Area of quadrilateral is used in the field of architecture, agriculture, design, and navigation also it helps to find distance between two points. It is required to find the area of buildings, park and other complexes.

How to calculate the area of a quadrilateral if one of its diagonals and both perpendiculars from the vertices are given?

When the diagonal(d) and the length of both perpendiculars (h, H) from the vertices are given, then the area of the quadrilateral is calculated by the formula:

Area of quadrilateral = (½) × d × (h + H)

What are the two main types of quadrilaterals?

The two main types of a quadrilateral are

• Regular Quadrilateral
• Irregular Quadrilateral

How to find the Area of a Quadrilateral using Heron’s Formula?

To find area of triangle using Heron’s Formula use the following steps:

• Step 1: Divide the quadrilateral in two triangles by joining its diagonal.
• Step 2: Find the area of both triangles using Heron’s formula.
• Step 3: Add both the areas to get the final answer.