Area of a Quadrilateral Formula

A quadrilateral is a plane figure which is bounded by four sides. Diagonals are the straight lines joining the opposite corners.  Diagonal is responsible to divide the quadrilateral into 2 triangles. Different types of quadrilateral are – square, rectangle etc. We will study the Area of a Quadrilateral formula in detail.

Area of a Quadrilateral Formula

What is a Quadrilateral?

A quadrilateral is a closed-end figure that has four sides. The interior angles add up to 360 degrees and opposite sides are parallel. The opposite angles are equal. Different formulas are in use to calculate the area of different forms of a quadrilateral. So let us take a closer look at the area of a quadrilateral formula.

Formulas of Quadrilateral

1] Area of Square

The area of a quadrilateral formula includes the formula for the area of a square. A square is a figure having four sides and all sides are equal. The facing sides are parallel and diagonals are also equal. The adjacent sides are at a right angle to each other.

Derivation:

Considering ABCD a square whose each side has length equal to ‘a’ and AC is a diagonal which divides the square ABCD into equal right triangles, named \Delta ABC &  \Delta ACD. Therefore,

 Area of the square ABCD = Area of \Delta ABC + Area of \Delta ACD

=\(\frac{1}{2}\left ( AB \right )\left ( BC \right )+ \frac{1}{2}\left ( AD \right )\left ( DC \right )\)

=\(\frac{1}{2}\left ( a \right )\left ( a \right )+\frac{1}{2}\left ( a \right )\left ( a \right )\)

=\(a^{2}\)

2] Area of Rectangle

A rectangle has four sides whose opposite sides are parallel and equal in length. Both the diagonals are equal and the angles between adjacent sides are at 90 degrees. Also, diagonals of a rectangle bisect each other.

Derivation:

Consider ABCD be a rectangle having side AB = a

and BC = b. The rectangle is divided into two right triangles by the diagonal AC, δABC and δADC . Area of rectangle ABCD = Area of δABC + Area of δADC

=  \(\frac{1}{2}\left ( AB \right )\left ( BC \right )+ \frac{1}{2}\left ( DC \right )\left ( AD \right )\)

= \(\frac{1}{2}ab+ \frac{1}{2}ab = ab\)

So, Area of rectangle ABCD = ab; Area = length x width

3] Area of Parallelogram

A parallelogram is a quadrilateral whose opposite sides are equal in length and parallel, but its diagonals are unequal and bisect each other.

Derivation:

Consider ABCD be a parallelogram whose base AB=a and Height DE=h. Then the area of parallelogram ABCD  = Area of rectangle DEFC

\(\left ( length \right )\left ( breadth \right )=\left ( EF \right )\left ( DE \right )\)

\(=\left ( DC \right )\left ( DE \right )\)

\(=\left ( AB \right )\left ( DE \right )= ab\)

4] Area of Rhombus

A quadrilateral having all sides equal with unequal diagonal, which bisect each other is. If a square is pressed from two opposite then rhombus is formed.

Here, AC = d1 and BD = d2 be the two diagonals.

As we see the diagonals of rhombus divide into four equal triangles.

So, \(Area of rhombus = 4\left ( area of one triangle \right )\)

\(=4\left ( \frac{1}{2}\times \frac{AC}{2}\times \frac{BD}{2} \right )\)

\(=\frac{AC\times BD}{2}\)

\(=\frac{d1\times d2}{2} = \frac{1}{2}\left ( product of two digonals \right )\)

5] Area of Trapezium

It is a quadrilateral whose two sides parallel and the other two are unparallel. The parallel sides of a trapezium are called the base.

Derivation:

Considering, ABCD is a trapezium whose sides AB and CD are parallel, AD and BC and unparallel sides. Say AB = a, CD = b, and DP = h BL= h, Since the diagonal BD divide the trapezium into two triangles ABD & BCD.

\(Area of Trapezium is= \Delta ABD + \Delta BCD\)

\(=\frac{1}{2}AB\times DP + \frac{1}{2}DC\times BL\)

\(=\frac{1}{2}ah+\frac{1}{2}bh\)

\(=\frac{1}{2}\left ( a+b \right )h\)

\(=\frac{Sum of parallel sides}{2}\times h\)

 Solved Examples

Q.1. Find the area of a parallelogram whose base is 24cm and height 13 cm respectively.

Ans- Here, b = 24cm, h = 13 cm

Area parallelogram =\(b\times h = 24 x 13 = 312 sq. cm\)

Q.2.Find the area of a trapezium whose parallel sides are 57 cm and 85 cm and perpendicular distance between them is 4cm.

Ans- Here, a = 85cm,  b = 57cm, h = 4cm

\(Area of Trapezium = \frac{a+b}{2}\times h\)

\(= \frac{85+57}{2}\times 4=142 \times 2= 284 sq.cm\)