The region that the quadrilateral’s four sides surround is known as its area. The term “region” is one that we are already familiar with. It is referred to as the area included within the perimeter of a flat object or figure. The conventional unit of measurement is square meters, and all measurements are made in square units. We are aware that the four-sided polygon is referred to as a quadrilateral.

What is Area of Quadrilateral?

Area of quadrilateral is the region that the quadrilateral’s four sides surround. A quadrilateral’s area is expressed in square units like m², in², cm² etc. There are various formulas for calculating the area of a regular quadrilateral. Numerous formulas, some of which are covered in the next section of this page, are used to determine the area of an irregular quadrilateral.

We determine the areas of each individual triangle to add up to the area of the quadrilateral ABCD. However, the height of a triangle must be determined in order to calculate its area.

Assume that the triangles ABD and BCD have heights of h1 and h2, respectively.

According to the above figure, the area of the quadrilateral ABCD = Area of △ABD + Area of △BCD

Area of the quadrilateral ABCD =

\( \left ( \frac{1}{2} \right )\times d\times h_{1} + \left ( \frac{1}{2} \right )\times d\times h_{2} = \frac{1}{2}\times d\times \left ( h_{1} +h_{2}\right )\) 

Thus, the following formula is used to determine a quadrilateral’s area:

Area of Quadrilateral formula = 1/2 x diagonal length x sum of the length of the perpendiculars drawn from the remaining two vertices.

\(= \left ( \frac{1}{2} \right )\times d\times \left ( h_{1}+h_{2} \right )\)

Formula for Area of Quadrilateral using Sides

Given the sides and two opposite angles of a quadrilateral, its area may be determined using Bretschneider’s formula.

Let’s take a look at a quadrilateral with the sides a, b, c, and d, and the opposite angles θ1 and θ2.

The quadrilateral’s area = \(\sqrt{\left ( s-a \right )\left ( s-b \right )\left ( s-d \right )-abcd \cos ^{2}\frac{\theta }{2}}\)

s is the quadrilateral’s semi-perimeter.

Area Formulas for Different Quadrilaterals 

1. Square

Area = side × side = a²

A square has all four sides equal and every angle is 90°.
To find the area, just multiply one side by itself.
For example, if one side is 5 cm, area = 5 × 5 = 25 cm².

2. Rectangle

Area = length × breadth = l × b

A rectangle has opposite sides equal and all angles are right angles.
To calculate the area, multiply the longer side (length) with the shorter side (breadth).
For example, if length is 8 cm and breadth is 3 cm, area = 8 × 3 = 24 cm².

3. Parallelogram

Area = base × height = b × h

A parallelogram looks like a tilted rectangle, with opposite sides equal.
To find its area, multiply the base with the vertical height (not the slant height).
If base = 10 cm and height = 4 cm, area = 10 × 4 = 40 cm².

4. Rhombus

Area = ½ × diagonal 1 × diagonal 2 = ½ × d₁ × d₂

A rhombus has all sides equal like a square, but angles are not 90°.
Its area is found using its two diagonals (lines joining opposite corners).
If diagonals are 6 cm and 8 cm, area = ½ × 6 × 8 = 24 cm².

 5. Kite

Area = ½ × diagonal 1 × diagonal 2 = ½ × p × q

A kite has two pairs of equal adjacent sides, and one diagonal cuts the other at a right angle.
Like the rhombus, its area is found by using the lengths of both diagonals.
If diagonals are 5 cm and 10 cm, area = ½ × 5 × 10 = 25 cm².

 6. Trapezium (or Trapezoid)

Area = ½ × (sum of parallel sides) × height = ½ × (a + b) × h

A trapezium has one pair of opposite sides that are parallel.
To get the area, add the lengths of the parallel sides, multiply by the height, and divide by 2.
If a = 7 cm, b = 5 cm, and height = 4 cm, area = ½ × (7 + 5) × 4 = 24 cm².

Area of Quadrilateral in Coordinate Geometry

When given coordinates, one method of determining the area is to use the distance formula to determine the lengths of the three sides. The Heron formula can then be used. However, this is a time-consuming process, especially if the side lengths are irrational.

Area of Quadrilateral in Coordinate Geometry Formula

Consider the ABCD quadrilateral that is shown below.

The vertices of the quadrilateral above are A (x1,y1),B (x2,y2),C (x3,y3)

We must now take the vertices of the quadrilateral ABCD in order to find its area
A (x1,y1), B (x2,y2)), C (x3,y3)

Write the letters of the quadrilateral ABCD column-wise as shown below. Arrange them in counterclockwise order.

The dark arrows represent the diagonal products \( x_{1}y_{2}, x_{2}y_{3}, x_{3}y_{4} and x_{4}y_{1}\)

\( x_{1}y_{2}+ x_{2}y_{3}+ x_{3}y_{4}+ x_{4}y_{1}\) ……(1)

The dotted arrows display the diagonal products \( x_{2}y_{1},x_{3}y_{2}, x_{4}y_{3}, x_{1}y_{4}\) respectively.

\( x_{2}y_{1}+ x_{3}y_{2}+ x_{4}y_{3}+ x_{1}y_{4}\) ……(2)

To calculate the area of the quadrilateral ABCD, subtract (2) from (1) and multiply the difference by 1/2.

Therefore, the quadrilateral ABCD’s area is= \( \left ( \frac{1}{2} \right ).\left \{ \left ( x_{1} y_{2}+x_{2}y_{3}+x_{3}y_{4}+x_{4}y_{1}\right )-\left ( x_{2}y_{1}+x_{3}y_{2}+x_{4}y_{3}+x_{1}y_{4}\right ) \right \}\)

Learn about Area of a Quadrilateral

Properties of a Quadrilateral 

A quadrilateral is a shape that has four sides, four corners (called vertices), and four angles. It is one of the most basic shapes in geometry.

One important rule about all quadrilaterals is that the sum of their interior angles is always 360 degrees. That means if you add all four angles inside the shape, the total will be 360°—no matter what the quadrilateral looks like.

In general, a quadrilateral can have sides of different lengths and angles of different sizes. These are called irregular quadrilaterals.

However, there are also special types of quadrilaterals that follow specific rules. For example:

• A square has all sides equal and all angles 90°.
• A rectangle has opposite sides equal and all angles 90°.
• A parallelogram has opposite sides that are equal and parallel, but the angles may not be 90°.

How to find Area of Quadrilateral in Coordinate Geometry

Steps to find the area of quadrilateral in coordinate geometry are as follows:

Example: Find the area of the quadrilateral whose coordinates are A(3,2), B(5,4), C(7,6), and D(5,4), in that sequence.

Solution: A quadrilateral’s area is the sum of the areas of the two triangles that result from dividing it in half.

Step: 1 Area of the ABCD-shaped quadrilateral = Area of △ABC + Area of △ACD

Step: 2 \( Area of \bigtriangleup ABC = \frac{1}{2}\left [ \left ( -3 \right )\left ( 4+6 \right )+5\left ( -6-2 \right )+7\left ( 2-4 \right ) \right ]\)

\( = \frac{1}{2}\left [ -30-40-14 \right ]\)

\( =\frac{1}{2} \left [ -84 \right ]\)

42 square unit

Step: 3 now, find the area of \( \bigtriangleup ACD = \frac{1}{2}\left [ -3\left ( -6+4 \right )+7\left ( -4-2 \right ) +\left ( -5 \right )\left ( 2+6 \right )\right ]\)

\( =\frac{1}{2}\left [ +6-42-40 \right ]\)

\( =\frac{1}{2}\left [ -76\right ]\)

38 square unit

Step: 4 Therefore, the ABCD quadrilateral’s area is 42 + 38 = 80 square units.

38 square unit

Step: 4 Therefore, the ABCD quadrilateral’s area is 42 + 38 = 80 square units.

Solved Examples on Area of Quadrilateral in Coordinate Geometry

Example 1:Determine the area of the quadrilateral with the following vertices: (7, 5), (4, 10), (-6, 11), and (-5, 2).

Solution:

Let the vertices of the quadrilateral ABCD be:

• A(7,5)
• B(4,10)
• C(−6,11)
• D(−5,2)D

We can apply the formula for the area of a quadrilateral given its vertices:

Area of the quadrilateral = (1/2) * [x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁ - (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁)]

Substitute the values into the formula:

Area = (1/2) * [(7 × 10) + (4 × 11) + (-6 × 2) + (-5 × 5) - (5 × 4) + (10 × -6) + (11 × -5) + (2 × 7)]

Simplifying the calculations:

Area = (1/2) * [70 + 44 - 12 - 25 - 20 - 60 - 55 + 14]

Area = (1/2) * [-44]

Area = 22 sq. units

Thus, the area of the quadrilateral is 22 sq. units.

Example 2: Given is a quadrilateral ABCD with heights of 8 cm for triangle ABD and 6 cm for triangle CBD and a diagonal of 20 cm. Calculate the ABCD quadrilateral’s area.

Solution: A = \( \frac{1}{2}\times d\times \left ( h_{1}+h_{2}\ \right )\)

d = 20 cm, h_{1} = 8 cm, h_{2} = 6 cm

\( = \frac{1}{2}\times 20\times \left ( 8+6 \right )cm^{2}\)

\( = \frac{1}{2}\times 20\times14cm^{2}\)

\( = 140 cm^{2}\)