Area of General Quadrilateral: Definition, Formula, Solved Examples

You have landed on the right page to learn about the Area of General Quadrilateral in detail. The amount of space or region bounded by a quadrilateral in two dimensions is called the area of the quadrilateral. A closed polygon bounded by four sides (line segments) is known as a quadrilateral. A quadrilateral has four sides, four vertices and four angles.

We have mainly six types of quadrilaterals that are classified based on the unique properties (sides, angles) of the quadrilateral. They are Trapezium, Parallelogram, Rectangle, Rhombus, Square and Kite. We have different methods to calculate the space occupied by the quadrilateral. We also have some formulas to find the area of the quadrilaterals.

This article will discuss how to find the area of the quadrilateral and formulas. Continue reading to know more.

What is a Quadrilateral?

A closed polygon bounded by four sides (line segments) is known as a quadrilateral. A quadrilateral has four sides, four vertices and four angles. The quadrilateral consists of only two diagonals. The sum of all angles in a quadrilateral (angle sum property of the quadrilateral) is equal to \({360^ \circ }\).

Quadrilaterals can be regular or irregular. A quadrilateral with all four sides equal is called a regular quadrilateral. On the other hand, quadrilaterals whose sides are unequal are known as irregular quadrilaterals.

We have mainly six types of quadrilaterals that are classified based on the unique properties (sides, angles) of the quadrilateral. They are

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

Here, a square and a rhombus are the regular quadrilaterals because the length of all sides is equal in them.

Area of Quadrilateral

The amount of space or region bounded by the quadrilateral in two dimensions is called the area of the quadrilateral. The perimeter gives the total length of the boundaries (sides) of the quadrilateral. The area of the quadrilateral is generally measured in square units like \({{\rm{m}}^{\rm{2}}}{\rm{,c}}{{\rm{m}}^{\rm{2}}}{\rm{,sq – km,sq – ft,sq – inc}}\) etc.

We have various methods of calculating the area of the quadrilateral. We also have several formulas to find the area of the quadrilaterals such as square, rectangle, parallelogram etc. Thus, the method of finding the area or the formula to be applied to find the area depends on the given information and type of the quadrilateral.

The area of the quadrilaterals like square, rectangle, rhombus etc., can be found directly by using the formulas. But the area of the general quadrilateral can be found by either splitting the quadrilateral into two triangles or using Heron’s formula, or Bretschneider′s formula, or using coordinate geometry, etc.

Learn About Properties of Quadrilaterals

Area of General Quadrilateral

One of the ways of finding the area of the general quadrilateral is by splitting the given quadrilateral into two triangles. To find the quadrilateral area in this method, we need the length of one diagonal and the heights of two triangles so formed.

Consider a quadrilateral \(ABCD.\)

Let \(“d”\) be the length of the diagonal \(BD.\) And, \(h_{1}\) and \(h_{2}\) be the heights of the triangles \(BCD\) and \(ABD,\) as shown in the figure.

The area of the quadrilateral is now obtained by adding the areas of the two triangles so formed.

The area of the quadrilateral \(ABCD=\) area of the triangle \(BCD+\) area of the triangle \(ABD.\)

\(\Rightarrow \operatorname{ar}(A B C D)=\frac{1}{2} \times d \times h_{1}+\frac{1}{2} \times d \times h_{2}\)

\(\Rightarrow \operatorname{ar}(A B C D)=\frac{1}{2} \times d \times\left(h_{1}+h_{2}\right)\)

Thus, the area of the quadrilateral when the length of one diagonal and heights are given is 

\(\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times length\, of\, the\, diagonal \times (sum\, of\, heights)}}\)

Area of Quadrilateral Using Heron’s Formula

Heron’s formula is used to find the area of the quadrilateral when the length of all four sides of the quadrilateral is given.

Consider the quadrilateral \(ABCD.\)

Divide the quadrilateral \(ABCD\) along with anyone diagonal, say \(BD.\)

The area of the quadrilateral is now obtained by adding the areas of the two triangles so formed.

Let \(AB=a, BC=b, CD=c, AD=d\) and \(BD=e.\) So, the semi-perimeter for \(\Delta BCD\) is \(s_{1}=\frac{b+c+e}{2}\) and that for \(\Delta ABD\) is \(s_{2}=\frac{a+d+e}{2}\)

The area of the quadrilateral \(ABCD=\) area of the triangle \(BCD+\) area of the triangle \(ABD.\)

\(=\sqrt{s_{1}\left(s_{1}-b\right)\left(s_{1}-c\right)\left(s_{1}-e\right)}+\sqrt{s_{2}\left(s_{1}-a\right)\left(s_{2}-d\right)\left(s_{2}-e\right)}\)

Area of Quadrilateral Using Sides and Opposite Angles

We can find the quadrilateral area when the lengths of sides and the values of opposite angles are given using the Bretschneider′s formula.

Let us consider the lengths of the sides of the quadrilateral are \(a, b, c, d\) and \(\theta_{1}, \theta_{2}\) are the two opposite angles.

Then the area of the quadrilateral can be found by using the Bretschneider′s formula as given below:

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

Here,

(i) \(\theta=\theta_{1}+\theta_{2}\) (Sum of given opposite angles)

(ii) \(s=\frac{a+b+c+d}{2}\) (Semi-perimeter of the quadrilateral

Area of Quadrilateral Using Coordinates

We can find the area of the quadrilateral when we have the coordinates of its vertices. 

Let us consider the quadrilateral \(ABCD,\) whose coordinates of the vertices are \(A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)\) and \(D\left(x_{4}, y_{4}\right)\) as shown in the cartesian plane.

Write the coordinates of the given quadrilaterals in the following manner.

Add the products of the elements of the diagonal as shown with the blue colour line.

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

Add the products of the elements of the diagonal as shown with the red colour line.

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

To find the area of the quadrilateral subtract equation (ii) from i And multiply the result by half.

\(\frac{1}{2}\left[\left(x_{1} y_{2}+x_{2} y_{3}+x_{3} y_{4}+x_{4} y_{1}\right)-\left(x_{2} y_{1}+x_{3} y_{2}+x_{4} y_{3}+x_{1} y_{4}\right)\right]\)

Note: We can find the area of the quadrilateral by splitting it into two triangles and adding their areas. Area of the triangle with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)\) is given by

\(\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]\)

Area of Quadrilateral Using Formulas

We have some set of formulas to calculate the area of the various types of the quadrilateral such as square, rectangle, rhombus, parallelogram, kite, trapezium and rectangle are listed below:

Solved Examples- Area of General Quadrilateral

Q.1. Asit draws a quadrilateral on the paper by taking the length of one diagonal is 10 cm, and the heights of the quadrilateral are 5 cm and 3 cm. Find the area of the quadrilateral.
Ans: Let the length of the diagonal \(A C=10 \mathrm{~cm} .\)
And, the height of the \(∆ADC\) is \(5 \mathrm{~cm}\), and the height of the \(\Delta A B C\) is \(3 \mathrm{~cm} .\)

Now, the area of the quadrilateral is \(\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times length\, of\, the\, diagonal \times (sum\, of\, heights)}}\)
\(=\frac{1}{2} \times 10 \times(3+5)\)
\(=10 \times 4=40 \mathrm{~cm}^{2}\)
Therefore, the area of the quadrilateral Asit drew is \(40 \mathrm{~cm}^{2}\)

Q.2. Sanju made the kite as shown below with the given measurements. Find the area of the kite made by Sanju.

Ans: Given Sanju made a kite \(PQRS.\) 
Area of the kite \(PQRS\) is equal to sum of the two parts, as shown in the figure.
In triangle \(PQS,\)
Semi perimeter \(\left(s_{1}\right)=\frac{4+4+5}{2}=\frac{13}{2}=6.5\) units
\(\operatorname{ar}(\Delta P Q S)=\sqrt{6.5(6.5-4)(6.5-4)(6.5-5)}=2.5 \sqrt{9.75}\) sq.units.
In triangle \(QRS,\)
Semi perimeter \((s)=\frac{6+6+5}{2}=\frac{17}{2}=8.5\) units
\(\operatorname{ar}(\Delta Q R S)=\sqrt{8.5(8.5-6)(8.5-6)(8.5-5)}=2.5 \sqrt{29.75}\) sq.units.
The area of the kite \(PQRS=\operatorname{ar}(\Delta P Q S)+\operatorname{ar}(\Delta Q R S)=2.5(\sqrt{9.75}+\sqrt{29.75})\) sq.units
Hence, the area of the kite made by Sanju is \(2.5(\sqrt{9.75}+\sqrt{29.75})\) sq.units.

Q.3. Find the area of the quadrilateral shown below:

Ans: Let, length of sides of the quadrilateral \(a=15, b=12, c=10, d=8\) and \({\theta _1} = {80^ \circ },{\theta _2} = {100^ \circ }\)
We know that area of the quadrilateral when the length of four sides and the two opposite angles are given by \(\sqrt{(s-a)(s-b)(s-c)(s-d)-a b c d \cos ^{2} \frac{\theta}{2}}\)
Semi perimeter \((s)=\frac{15+12+10+8}{2}=\frac{45}{2}=22.5\)
And, \(\theta = {100^ \circ } + {80^ \circ } = {180^ \circ }\)
The area of the quadrilateral is 
\( = \sqrt {22.5(22.5 – 15)(22.5 – 12)(22.5 – 10)(22.5 – 8) – 15 \times 12 \times 8 \times 10 \times \frac{{{{180}^ \circ }}}{2}} \)
\(= \sqrt {22.5 \times 7.5 \times 10.5 \times 12.5 \times 14.5 – 15 \times 12 \times 8 \times 10 \times {{90}^ \circ }} \)
\(=\sqrt{321152.3438-0}\)
\(=566.7\) (Approx)
Hence, the area of the quadrilateral is \(566.7\) sq units (approximately).

Q.4. A swimming pool has an area of \(100 \mathrm{~m}^{2}\) which is in the shape of a square. Find the length of the side of the swimming pool.
Ans: Given a swimming pool is in the shape of a square.
Given, area of the swimming pool (square) is \({\rm{100}}{{\rm{m}}^{\rm{2}}}\)
We know that area of the square is \({\rm{sid}}{{\rm{e}}^{\rm{2}}}{\rm{.}}\)
\(\Rightarrow {\rm{sid}}{{\rm{e}}^2} = 100\;{{\rm{m}}^2}\)
Applying square root on both sides of the equation
\(\Rightarrow {\rm{side = }}\sqrt {{\rm{100}}} \)
\(\Rightarrow {\rm{side}} = 10\;{\rm{m}}\)
Hence, the length of the side of the swimming pool is \(10 \mathrm{~m}\)

Q.5. The length and width of the rectangular farm are 80 yards and 60 yards. Find the area of the farm.

Ans: Given: Length of the rectangle \((l)=80\) yds.
Width of the farm \((b)=60\) yds.
We know that area of a rectangle is \({\rm{length \times breadth}}{\rm{.}}\)
So, the area of the farm is \(=80 \times 60=4800 \mathrm{sq} \cdot \mathrm{yd}\)

Summary

In this article, we have discussed the definitions of the quadrilateral and the area of the quadrilateral. We have studied the quadrilateral and its types also the basic properties of the quadrilateral.

This article gives the methods of finding the area of the quadrilateral, such as finding the area by using Heron’s formula, finding the area by using Bretschneider′s formula, finding the area of the quadrilateral when the diagonal and heights are given and also finding the area of the quadrilateral in the cartesian plane when its coordinates are given. This article also gives the solved examples, which helps to find the area of the quadrilateral easily.

Learn About Area of Quadrilaterals

Frequently Asked Questions (FAQs)

Q.1. What is the area of a quadrilateral?
Ans: The amount of space or region bounded by the quadrilateral in two dimensions is called the area of the quadrilateral.

Q.2. What is the formula of area of the general quadrilateral?
Ans: The area of the general quadrilateral when the length of one diagonal and heights are given is \(frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times length\,of\,the\,diagonal \times (sum\,of\, heights)}}{\rm{.}}\)

Q.3. What is the Bretschneider′s formula ?
Ans: The Bretschneider′s formula is used to calculate the area of the quadrilateral. The area of the quadrilateral when the sum of four sides and two opposite angles is given below:
\(\sqrt{(s-a)(s-b)(s-c)(s-d)-a b c d \cos ^{2} \frac{\theta}{2}}\)
Here \(s=\frac{a+b+c+d}{2}\) and \(\theta=\) of opposite angles.

Q.4. How do you use Heron’s formula to find the area of the quadrilateral?
Ans: To find the quadrilateral area, divide the given quadrilateral into two triangles along any diagonal. The area of each triangle can be found by using Hern’s formula.
The Heron’s formula of finding the area of the triangle with sides \(a, b, c\) is given by 
\(\sqrt{s(s-a)(s-b)(s-c)}, s=\frac{a+b+c}{2}\)

Q.5. What is the area of the quadrilateral when the coordinates of the vertices are given?Ans: The area of the quadrilateral with vertices \(A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)\) and \(D\left(x_{4}, y_{4}\right)\) is \(\frac{1}{2}\left[\left(x_{1} y_{2}+x_{2} y_{3}+x_{3} y_{4}+x_{4} y_{1}\right)-\left(x_{2} y_{1}+x_{3} y_{2}+x_{4} y_{3}+x_{1} y_{4}\right)\right]\)

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