Area of Quadrilateral - Introduction, Formulae, Calculations

A quadrilateral is a polygon with four sides. A quadrilateral is a closed, two-dimensional shape made up of four points, three of which are non-collinear points. There are four vertices, four angles, and four sides in a quadrilateral. There are two types of quadrilaterals, regular quadrilaterals and irregular quadrilaterals. If all four sides have the same length, the quadrilateral is said to be regular. Quadrilaterals having unequal sides are referred to as irregular quadrilaterals. Square, rhombus, rectangle, etc are some examples of quadrilaterals.

Any region that falls within a specific border or figure is referred to as an area. The area contained by a figure with four sides is referred to as the area of a quadrilateral. A quadrilateral's area is expressed in square units. Usually, square meters is used as the standard measurement unit for areas.

Area Of Quadrilateral By Dividing Into Two Triangles

Let ABCD be a quadrilateral with AC as one of its diagonals. ABC and ADC are triangles with height lengths equal to h1 and h2 respectively, considering AC as the base with length ‘b’.

The area of the quadrilateral ABCD is equal to the sum of areas of triangle ABC and triangle ADC.

Area of triangle ABC = \frac{1}{2} \times h_1 \times b

Area of triangle ADC = \frac{1}{2} \times h_2 \times b

Area of quadrilateral = Area of triangle ABC + Area of triangle ADC

= (\frac{1}{2} \times h_1 \times b) + (\frac{1}{2} \times h_2 \times b)

= \frac{1}{2} \times b\times (h_1+h_2)

Area Of Quadrilateral Using Its Sides

The Bretschneider's formula can be used to calculate a quadrilateral's area given its sides and two of its opposite angles. Consider a quadrilateral with the sides a, b, c, and d, with the opposite angles \theta_1 and \theta_2 on it. Then the area of the quadrilateral is given by:

\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2{(\theta/2)}}

Where,

s is the semiperimeter of the quadrilateral.

s=\frac{a+b+c+d}{2}

and,

\theta=\theta_1+\theta_2

Area Of Quadrilateral Using Heron’s Formula

In order to use this method, we first divide the given quadrilateral into two triangles, and then we use Heron's formula to calculate the areas of each triangle individually. The area of both triangles is added up in order to determine the total area of the quadrilateral.

Let us consider a triangle with side lengths equal to a, b and c then the area of the triangle using Heron’s formula is given by:

Area= \sqrt {s (s-a) (s-b) (s-c)}

where,

s is the semi-perimeter of the triangle.

s=\frac{a+b+c}{2}

Area Of A Quadrilateral With Given Coordinates

Let A(x_1, y_1) , B(x_2, y_2) , C(x_3, y_3) and D(x_4, y_4) be the coordinates of vertices of a quadrilateral in a coordinate plane. To determine the area of the quadrilateral, we write the vertices as:

\frac{1}{2}\begin{bmatrix}

x_1 & x_2 & x_3 & x_4 & x_1\\

y_1 & y_2 & y_3 & y_4 & y_1

\end{bmatrix}

Add the diagonal products x_1y_2 , x_2y_3 , x_3y_4 and x_4y_1.

Subtract the diagonal products x_2y_1, x_3y_2, x_4y_3 and x_1y_4.

So, the area is given by:

Area = (x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1)-(x_2y_1 + x_3y_2 + x_4y_3 + x_1y_4)

Area Of Some Important Quadrilaterals

• Area of a square with a side length equal to ‘a’ is equal to a^2 .

• Area of a rectangle with length and breadth equal to ‘a’ and ‘b’ respectively is equal to ab.

• Area of a parallelogram with base ‘b’ and height ‘h’ is equal to bh.

• Area of a trapezoid with the length of parallel sides equal to ‘a’ and ‘b’ and height ‘h’ is equal to \frac{1}{2}(a+b)h

• Area of a rhombus with length of diagonals equal to d_1 and d_2 is equal to \frac{1}{2}\times d_1 \times d_2 .

• Area of a kite with length of diagonals equal to d_1 and d_2 is equal to \frac{1}{2}\times d_1 \times d_2 .

Examples

1. Find the area of a square whose side length is equal to 4cm.

Solution: Area of a square is equal to a^2

Where, a is the side of the square.

Given, a = 4cm

Area = 4^2

=16 sq. cm

2. Find the area of a rectangle whose length is 5in and width is 4in.

Solution: Area of a rectangle with length ‘a’ and width ‘b’ is equal to ab.

Given a = 5in

and

b = 4in

Area = 20 sq. in

3. Find the area of a kite whose diagonal lengths are 20m and 10m.

Solution: Area of a kite with length of diagonals equal to d_1 and d_2 is equal to \frac{1}{2}\times d_1 \times d_2 .

Given,

d_1=20\\

d_2=10\\

Area=\frac{1}{2}\times 20 \times 10\\

=100m^2