How to Calculate the Area of Quadrilaterals?

Before carrying on with the area of the quadrilaterals let’s recall the base figure or shape and its properties. Quadrilaterals are four-sided planes.

1. ‘Quad’= four, ‘Lateral’=side
2. Formed by joining four points
3. The points are non-collinear
4. The quadrilateral is a closed figure
5. Diagonal divide the quadrilateral into two triangles
6. The sum of angles of a quadrilateral is 360 degrees
7. A quadrilateral can be regular or irregular.

Quadrilateral family

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

Area of quadrilateral is the amount of space that is present in it. The area is always measured in square units such as m^2, cm^2, inch^2, etc.

The process of finding the area of a quadrilateral depends on the information available and the type of the quadrilateral.

Areas of quadrilaterals specifically

1. Area of parallelogram: The area of a parallelogram is the product of the length of one side and the perpendicular distance between the parallel sides of the parallelogram. AREA = BASE*HEIGHT
2. Area of rhombus: The area of rhombus is equal to half the product of diagonals. AREA = 1/2*D1*D2, where D1 and D2 are the diagonals of rhombus
3. Area of trapezium: The area of a trapezium is equal to half the product of the sum of parallel sides and the distance between them. AREA=1/2*HEIGHT*SUM OF PARALLEL SIDES
4. Area of rectangle: The area of rectangle is the product of length and breadth. AREA = LENGTH*BREADTH
5. Area of square: The area of square is the square of side. AREA=SIDE^2
6. Area of kite: The area of kite is equal to half of the product of its diagonals. AREA OF KITE = 1/2*d1*d2 , where d1 and d2 are the diagonals of the kite.

If the quadrilateral does not belong to any of the specific categories of the quadrilaterals that are mentioned above then we can find the area of the quadrilateral by Bretschneider’s formula or by dividing the figure into two triangles.

This is efficiently explained by the CUEMATH team of professionals.

.

Now let’s learn how to calculate the area of a quadrilateral that does not seem to be a member of the quadrilateral family.

Area of quadrilateral

Dividing into two triangles, Area= ½*d*(h1+h2)

Bretschneider’s formula, Area= √(s-a)(s-b)(s-c)(s-d)-abcd cos^ϴ/2

Here : s=(a+b+c+d)/2, where s is the semi perimeter of quadrilateral

ϴ=ϴ1+ϴ2

Area of quadrilateral by dividing it in two triangles: To find an area in such figures we should be provided with the heights (perpendicular distance of the diagonal from two opposite vertices) of the triangles formed by the diagonal.

Areas of triangles formed by dividing the quadrilateral = ½*d*h1 ; ½*d*h2, where h1, h2 are the heights of the triangles.

Thus Area of quadrilateral=(1/2*d*h1)+(1/2*d*h2)=1/2*diagonal*sum of heights

Or we can also find the area using heron’s formula,

The area of triangles with 3 sides a, b, c = √s(s-a)(s-b)(s-c), where s is the semi-perimeter of the triangle i.e. s= (a+b+c)/2

Area of quadrilateral using Coordinates concept

The area of a quadrilateral can be calculated when coordinates of the vertices are known.

Area = 1/2{(x1y2+x2y3+x3y4+x4y1)-(x2y1+x3y2+x4y3+x1y4)}

Cyclic quadrilaterals

A quadrilateral whose all four vertices lie on a circle is called a cyclic quadrilateral. This is also sometimes known as an inscribed quadrilateral. The definition states that a quadrilateral that is circumscribed in a circle is a cyclic quadrilateral. The sum of the opposite angles of a cyclic quadrilateral is 180 degrees.

If a, b, c, d are the sides of the inscribed quadrilateral then its area is given by:

[latex]Area= √{(s-a)(s-b)(s-c)(s-d)}[/latex]

Where s is the semi-perimeter

S=1/2(a+b+c+d)

Cyclic quadrilaterals theorems

1. In a cyclic quadrilateral, the sum of either pair of opposite angles is supplementary.
2. The ratio between diagonals and sides can be defined and is known as Cyclic quadrilateral theorem. If there is a quadrilateral which is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.

These two theorems may be helpful in the calculation of the area of cyclic quadrilaterals.