Area of Quadrilateral - Introduction, Formulae, Calculations

Quadrilaterals are four-sided polygon with four vertices and four angles adding up to 360 degrees. Area of quadrilateral is defined as the region covered by all of its sides. We already have an understanding of the concept of area which is defined as the region coved by a 2-D figure, the units used being square uints. We know that a polygon with four sides is called a quadrilateral and it can be a square, rectangle, rhombus, kite, parallelogram and trapezium. In this article we will learn about what is the area of quadrilateral, area of quadrilateral formula in coordinate geometry, etc and much more.

This Story also Contains

1. What Is the Area of Quadrilateral?
2. Area of Quadrilateral Formula using Different Conditions
3. Area Formulas of Different Types of Quadrilaterals
4. Area of Quadrilateral Formula in Coordinate Geometry
5. Area of Quadrilateral in Vector Form
6. Area of Quadrilateral Examples

What Is the Area of Quadrilateral?

A Quadrilateral is a four sided polygon with four vertices and four angles adding up to 360 degrees. The area of quadrilateral is defined as the space covered by all the sides of a quadrilateral in an xy plane or a 2-dimensional space. The process of finding the area of a quadrilateral depends upon its type and the information available about the quadrilateral to us. Since we know that area is always represented in sq units like sq m, sq cm, we use the same units when we write the final expression for area of quadrilateral.

Area of Quadrilateral Formula

The formula of area of quadrilateral can be found out using different methods such as dividing the quadrilateral into two triangles, or by using Heron’s formula or by using sides of the quadrilateral. So depending upon what values are provided to us in the question, we make use of below derived formuas to make our calculations less complex and efficient. Let us now have a look on each of them one by one.

How to find Area of Quadrilateral by Dividing Into Two Triangles?

Consider a quadrilateral ABCD in which the length of the diagonal BD is known to be ' $d$ '. ABCD can be divided into two triangles by the diagonal BD. To find its area, we should be knowing the heights of the triangles ABD and BCD. Let us assume that the heights of the triangles ABD and BCD are given to be $h_1$ and $h_2$ respectively. We will find the area of the quadrilateral ABCD by adding the areas of triangles ABD and BCD.

Here, the area of the triangle ABD $=(\frac{1}{2}) \times \mathrm{d} \times \mathrm{h}_1$.
The area of the triangle BCD $=(\frac{1}{2}) \times \mathrm{d} \times \mathrm{h}_2$.
From the above figure, the area of quadrilateral $ABCD=$ area of $\triangle ABD+$ area of $\triangle BCD$.
Thus, the area of quadrilateral ABCD $=(\frac{1}{2}) \times d \times h_1+(\frac{1}{2}) \times d \times h_2=(\frac{1}{2}) \times d \times(h_1+h_2)$.
Thus, the formula used to find the area of quadrilateral when one of its diagonals and the heights of the triangles (formed by the given diagonal) are given is,

Formula for area of quadrilateral $=(\frac{1}{2}) \times$ Diagonal $\times($ Sum of heights $)$

Area of Quadrilateral Formula using Different Conditions

The area of quadrilateral formula using different conditions like area of quadrilateral formula with sides given and area of quadrilateral using Heron's formula.

Area of Quadrilateral Formula with Sides Given

When we are given sides and two of the opposite angles, we can find its area using the Bretschneider′s formula. Let us consider a quadrilateral whose sides are e, f, g, and h, and two of its opposite angles are θ1 and θ2 as shown in the diagram below:

Then the formula for area of quadrilateral $=\sqrt{ (s-e)(s-f)(s-g)(s-h)-e f g h \cos^2 \frac{\theta}{2} }$ where
- $s=$ semi-perimeter of the quadrilateral $=\frac{(e+f+g+h) }{2}$
- $\theta=\theta 1+\theta 2$

Area of Quadrilateral Using Heron's Formula

We already know that the area of triangle with sides $x, y$, and $z$ is $\sqrt{ }(\mathrm{s}-\mathrm{x})(\mathrm{s}-\mathrm{y})(\mathrm{s}-\mathrm{z})$, according to the famous herons formula application, where ' s ' is the semi-perimeter of the triangle, i.e., $s=\frac{(x+y+z)}{2}$. Hence, to find the Area of Quadrilateral Using Heron's Formula, We follow the below steps:
- We initially divide it into two triangles using a diagonal whose length is known to us as it will be given in the question.
- Next we apply heron's formula to each of the triangle individually for finding its area.
- At last, we add the areas of two triangles which gives the formula for area of quadrilateral.

Area Formulas of Different Types of Quadrilaterals

The area formulas of different types of quadrilaterals include the area of square, rectangle, parallelogram, trapezium, rhombus and kite shape.

Area of quadrilateral formula class 8

We know that there are 6 types of quadrilaterals, namely, rectangle, square, parallelogram, rhombus, trapezoid and kite. All the figures have their own special formulas for areas which are widely used in our calculations involving geometry and also in real lfe situations of building construction, architecture, geometric shapes and figures, etc. Now we look at formulas to find the area of different shapes.

Now, let us look into the area of quadrilateral in coordinate geometry.

Area of Quadrilateral Formula in Coordinate Geometry

In the quadrilateral given above, $\mathrm{P}(\mathrm{x} 1, \mathrm{y} 1), \mathrm{Q}(\mathrm{x} 2, \mathrm{y} 2), \mathrm{R}(\mathrm{x} 3, \mathrm{y} 3)$ and $\mathrm{S}(\mathrm{x} 4, \mathrm{y} 4)$ are the vertices.
To find area of quadrilateral PQRS, we take the vertices $P(x 1, y 1), Q(x 2, y 2), R(x 3, y 3)$ and $\mathrm{S}(\mathrm{x} 4, \mathrm{y} 4)$ of the quadrilateral PQRS and write them as shown below,

Now, we add the diagonal products $\mathrm{x} 1 \mathrm{y} 2, \mathrm{x} 2 \mathrm{y} 3, \mathrm{x} 3 \mathrm{y} 4$ and x 4 y 1 that are shown by the blue arrows in the above image.

$
(x 1 y 2+x 2 y 3+x 3 y 4+x 4 y 1) \rightarrow(1)
$

Next, we add the diagonal products $\mathrm{x} 2 \mathrm{y} 1, \mathrm{x} 3 \mathrm{y} 2, \mathrm{x} 4 \mathrm{y} 3$ and x 1 y 4 that are shown by the orange arrows.

$
(x 2 y 1+x 3 y 2+x 4 y 3+x 1 y 4) \rightarrow(2)
$

We subtract (2) from (1) and multiply the difference by $\frac{1}{2}$ to get area of quadrilateral PQRS.
So, area of quadrilateral formula coordinate geometry is given as,

$
A=(\frac{1}{2}) \cdot\{(x 1 y 2+x 2 y 3+x 3 y 4+x 4 y 1)-(x 2 y 1+x 3 y 2+x 4 y 3+x 1 y 4)\}
$

Area of Quadrilateral in Vector Form

The quadrilateral can be divited into two triangles. For example, consider a quadrilateral ABCD. If te quadrilateral is divided into two triangles $\Delta ABC$ and $\Delta ACD$, Then the area of quadrilateral in vector form is equal to the sum of the area of the triangles $\Delta ABC$ and $\Delta ACD$.

The area of triangle $\Delta ABC$ in vector form $= \frac{1}{2} \times|\overrightarrow{A B} \times \overrightarrow{A C}|$

The area of triangle $\Delta ABC$ in vector form $= \frac{1}{2} \times|\overrightarrow{A C} \times \overrightarrow{A D}|$

Therefore, the area of quadrilateral in vector form is $\frac{1}{2} \times|\overrightarrow{A B} \times \overrightarrow{A C}| + \frac{1}{2} \times|\overrightarrow{A C} \times \overrightarrow{A D}|$

Area of Quadrilateral Examples

Example 1: Find the area of rectangle whose length is 20 in and width is 15 in.
Solution: The length of the rectangle is, I $=20$ in.
Its breadth is, $\mathrm{b}=15$ in.
Using the formulas of the area of a quadrilateral, the area (A) of the given rectangle is,

$
A=1 \times b=20 \times 15=300 \mathrm{in}^2
$

Example 2: Find the area of kite, diagonals are 16 units and 12 units.
Solution: The diagonals of kite are, $\mathrm{d} 1=16$ units and $\mathrm{d} 2=12$ units.
The area (A) of the given kite is, $A=(\frac{1}{2}) \times d 1 \times d 2=(\frac{1}{2}) \times 16 \times 12=96$ square units.

Example 3: Calculate area of quadrilateral shown below.

Solution: The sides of quadrilateral are,

$
p=10 ; q=12 ; r=9 ; \text { and } s=10
$

Its semi-perimeter is, $s=\frac{(p+q+r+s)}{2}=\frac{(10+12+9+10)}{2}=20.5$.
Sum of angles, $\theta=100^{\circ}+80^{\circ}=180^{\circ}$.
The area $(A)$ of the given quadrilateral is found using the Bretschneider's formula.

$
\begin{aligned}
& A=\sqrt{(s-p)(s-q)(s-r)(s-s)-a b c d \cos^2 (\frac{\theta}{2})} \\
& A=\sqrt{ }(20.5-10)(20.5-12)(20.5-9)(20.5-10)-(10 \cdot 12 \cdot 9 \cdot 10) \cos ^2 (\frac{180}{ 2})=
\end{aligned}
$

Example 4: Calculate the area of the quadrilateral formed with the vertices $(-4,2),(2,4)$, $(8,-6)$ and $(-5,-4)$.

Solution: Let $P(-4,2), Q(2,4), R(8,-6)$ and $S(-5,-4)$ be the vertices of a quadrilateral $P Q R S$.

$
\begin{aligned}
& \mathrm{P}(-4,2)=\left(\mathrm{x}_1, \mathrm{y}_1\right) \\
& \mathrm{Q}(2,4)=\left(\mathrm{x}_2, \mathrm{y}_2\right) \\
& \mathrm{R}(8,-6)=\left(\mathrm{x}_3, \mathrm{y}_3\right) \\
& \mathrm{S}(-5,-4)=\left(\mathrm{x}_4, \mathrm{y}_4\right)
\end{aligned}
$

We know that,
Area of quadrilateral PQRS $=(\frac{1}{2}) \cdot\left[\left(x_1 y_2+x_2 y_3+x_3 y_4+x_4 y_1\right)-\left(x_2 y_1+x_3 y_2+x_4 y_3+x_1 y_4\right)\right]$

Substituting the values,

$
\begin{aligned}
& =(\frac{1}{2}) \cdot\{[-4(4)+2(-6)+8(-4)+(-5) 2]-\{[2(2)+8(4)+(-5)(-6)+(-4)(-4)]\} \\
& =(\frac{1}{2}) \cdot[(-16-12-32-10)-(4+32+30+16)] \\
& =(\frac{1}{2})[-70-82] \\
& = \frac{152}{2} \text { \{since area cannot be negative }\} \\
& =76 \text { sq units. }
\end{aligned}
$

Example 5: Find area of quadrilateral with diagonal as 20 and $\mathrm{h} 1=7 \mathrm{~cm}, \mathrm{~h} 2=5 \mathrm{~cm}$ ?
Solution: Formula of area of quadrilateral $=(\frac{1}{2}) \times$ Diagonal $\times($ Sum of heights $)= \frac{1}{2} \times 20 \times$ $12=120$ sq units.

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