Area of quadrilateral – Derivation, Formulas and Examples

What is a quadrilateral?

A quadrilateral is a 2-dimensional shape with four sides. Quad signifies four, and therefore the word ‘quadrilateral’ is given to a solid shape having four sides. There are numerous types of quadrilaterals, each with its unique characteristics and area formulas. 

The quadrilaterals are of two different categories — regular and irregular quadrilateral. Regular quadrilaterals have all sides with equal length, whereas an irregular quadrilateral has sides of unequal length.

Types of quadrilaterals

Quadrilaterals are of six different types with differing properties. The only common thing between these six quadrilaterals is that they have four sides and four angles. 

The name of these six types of quadrilaterals are as follows:

1. Kite
2. Rectangle
3. Square
4. Parallelogram
5. Rhombus
6. Trapezium

Out of these six quadrilaterals, the square is the only regular quadrilateral, whereas kite, rectangle, parallelogram, rhombus, trapezium are irregular quadrilaterals.

General properties of quadrilaterals

• The sum of a quadrilateral’s inner angles is 360°.
• There are four sides, four angles, and four vertexes on each quadrilateral.
• Two sets of adjacent quadrilateral angles have a sum of 180°.

What is the area of a quadrilateral?

The area of a quadrilateral such as square, rectangle, parallelogram refers to the enclosed region within its sides. We measure it in square units. The calculation of a quadrilateral’s area depends on its type and nature of the information given. You can calculate the area after dividing the quadrilateral into two triangles. Then, after calculating the area of these two triangles, you can add both to find the area of the quadrilateral. 

Deriving the area formula by dividing the quadrilateral into two triangles

Consider the quadrilateral ABCD with a diagonal ‘d’ connecting vertices B and D. The diagonal ‘d’ divides this quadrilateral into two triangles: BCD and ABD. Now, to find the area of these two triangles, we must know their height. Assume that the height of these triangles is h_1 \text{ and }h_2. 

Area of triangle BCD = \frac{1}{2} \times d \times h_1

Area of triangle ABD = \frac{1}{2} \times d \times h_2

Now, area of quadrilateral ABCD = area of triangle BCD + area of triangle ABD

Area of quadrilateral ABCD = (\frac{1}{2} \times d \times h_1)+(\frac{1}{2} \times d \times h_2)

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

Therefore, area of quadrilateral ABCD = \frac{1}{2} x diagonal x (sum of heights of each triangle)

Deriving the area formula by using Heron’s formula

We have seen that you can calculate the area of a quadrilateral by dividing it into two triangles. But, what will you do if you do not know the heights of the triangle?

If you have the measurement of each side of the triangle, you can use Heron’s formula to find the area of these triangles and add them up to obtain the area of a quadrilateral. 

According to Heron’s formula, the area of a triangle with sides a, b, and c is 

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

Here, ‘s’ is the semi-perimeter, i.e., s=\frac{(a+b+c)}{2}.

Upon finding the area of each triangle through Heron’s formula, add the areas of the two triangles, and you will obtain the area of a quadrilateral.

Area formulas for different types of quadrilaterals

While the derivations mentioned above will turn out to be fun for you, to make it easier, we have specific formulas to find the area of the six different types of quadrilaterals.

Solved examples 

Question 1: The length of the parallel sides of a trapezium is 12 feet and 16 feet. Calculate the area of the trapezium if its height is 6 feet. 

Ans:

Length of AC = 12 feet

Length of BD = 16 feet

Height of trapezium = 6 feet

Therefore, area of trapezium 

= ½ x (AC+BD) x height

= ½ x 12 x 16 x 6 \text { feet² }

= 576 \text { feet² }

Hence, the area of this trapezium is 576 \text { feet² }.

Question 2: In the parallelogram PQRS, the length of its parallel sides is 26 cm, and the area is 338 cm². What is the measure of the height of this parallelogram?

Ans:

Base of Parallelogram = 26 cm

Area = 338 cm²

Since the area of parallelogram = base x height

We can say that the height of the parallelogram 

=\frac{\text { area of parallelogram }}{\text { base of parallelogram }}

Height of parallelogram = \frac{338 \mathrm{~cm}^{2}}{26 \mathrm{~cm}}

Therefore, the height of parallelogram PQRS = 13 cm.