Area of a Quadrilateral – Types, Formulas, Properties & Solved Examples

In geometry, a quadrilateral is a flat shape with four sides, four corners (called vertices), and four angles. It is formed by joining four points, making sure that no three points lie on the same straight line. Some common types of quadrilaterals are rectangles, squares, trapeziums, and parallelograms.

The area of a quadrilateral is the amount of space it covers on a flat surface. This is measured in square units such as square meters (m²), square centimeters (cm²), etc.

To find the area of a quadrilateral, the method you use depends on the type of quadrilateral and the information given. If the shape is irregular and doesn’t match any standard type, we can either:

1. Divide it into two triangles and find the total area, or
2. Use a special formula called Bretschneider’s formula, which works when all four sides and two opposite angles are known.

What is Area of a Quadrilateral?

The area of a quadrilateral is the space or region that is enclosed within its four sides. It tells us how much surface the shape covers and is measured in square units like cm² or m².

In general, for many quadrilaterals, the area can be found by multiplying the base by the height, especially when the shape has straight and parallel sides like in rectangles and parallelograms.

There are also special formulas used to calculate the area of different shapes such as triangles, squares, circles, and different types of quadrilaterals.

Quadrilaterals are mainly divided into two types:

1. Regular Quadrilaterals – All sides and angles are equal, like a square.
2. Irregular Quadrilaterals – Sides and angles are not all equal, like a trapezium or any random four-sided shape.

We can find the area of a quadrilateral using the following ways:

1. Area of a Quadrilateral by Splitting it into 2 Triangles
2. Area of a Quadrilateral Using Sides
3. Area of a Quadrilateral Using Heron’s Formula

Area of a Quadrilateral by Splitting it into 2 Triangles

Area of a Quadrilateral by dividing it into 2 triangles can be obtained as follows:

Consider a Quadrilateral PQRS as shown above.

Divide it into two triangles where are the heights of triangles PSR and PQR respectively as shown in the figure below:

Area of triangle PSR + Area of triangle PQR =

Therefore the area of a Quadrilateral PQRS=Area of Triangle PSR+Area of Triangle PQR.

Here d=PR=diagonal

Area of a general Quadrilateral

Consider the above quadrilateral PQRS, here the diagonal=15cm and the heights of the triangles PSR and PQR are 5 cm and 7 cm, respectively. Find the area of the quadrilateral PQRS.

Area of a general Quadrilateral.

.

Area of Quadrilateral using Sides

If we consider the sides of a quadrilateral as a, b, c, d are given, and two of its opposite angles are specified as. Then the area of the quadrilateral can be computed by the below formula:

Area of the quadrilateral: .

where = a, b,c,d are the sides of a quadrilateral and is the opposite angles.

s = semi-perimeter of the quadrilateral and.

Such a quadrilateral, where the area is obtained using the sides of a quadrilateral and 2 of its opposite angles, is also known as Bretschneider′s formula.

Area of Quadrilateral using Heron’s Formula

Another approach to calculating the area of a quadrilateral is employing Heron’s formula. In general, we apply this formula to obtain the area of a triangle when the three sides are different. The below-mentioned procedure is used to locate the area of the quadrilateral.

• The first thing is to split the quadrilateral into two triangles operating on a diagonal. Here the diagonal length is known to us.
• In the next step, apply Heron’s formula separately for both the newly generated triangles to find the area of a quadrilateral with the below formula.
• Here a, b, and c are the sides of the obtained triangle.
• = semi-perimeter of the triangle.
• Lastly, add the area of two triangles to obtain the area of the complete quadrilateral.

Area of Quadrilateral with Vertices

To find the area of a quadrilateral with vertices, you need to know the coordinates of the four vertices. Once you have the coordinates, you can use the formula for the area of a quadrilateral to calculate the area.

Let , , and be the vertices of a quadrilateral as shown below.

Learn about Construction of Rhombus

There are 2 methods for determining the area of the quadrilateral at hand.

Method 1 for finding Area of Quadrilateral with Vertices

In order to calculate the area of quadrilateral , we must first select the vertices , , and in a specific order (counterclockwise), and then arrange them in a column-wise format as demonstrated below.

 Observe the directions given in the dark arrows, add the diagonal products, i.e., and .

….(1)

Now, consider the dotted arrows and add the diagonal products, i.e., and .

—(2)

Subtract (2) from (1) and multiply the difference by .

Therefore, the formula for the area of quadrilateral using vertices is:

.

Method 2 for finding Area of Quadrilateral with Vertices

In this method, we need to divide the given quadrilateral into two triangles. Then, find the area of each triangle and add it up to get the area of the quadrilateral.

Area of quadrilateral = Area of triangle + Area of triangle .

Note: Area of triangle with vertices , and is given by:

.

Example: Consider the following four points: , , , . These four points are the vertices of a quadrilateral:

In this scenario, we will partition the quadrilateral into two triangles by utilizing one of the diagonals. Then, we will ascertain the area of each triangle (ensuring it is a positive value) and sum them together to acquire the total area. In the given illustration, the quadrilateral has been separated into two triangles, and as shown below.

Now, we separately calculate the areas of the two triangles.

Area of :

sq. units.

Area of :

sq. units

So, area of quadrilateral = Area of triangle + Area of triangle = sq. units.

Formulas for Area of a Quadrilateral

Quadrilateral is derived from a Latin word where ‘Quadra’ indicates four and ‘Latus’ means sides. All four sides of a quadrilateral may or may not be identical.

All the formulas through which we can obtain the area of a quadrilateral are:

Area of Quadrilateral	Area of Quadrilateral Formula
General Formula Obtained by dividing a quadrilateral into two triangles.	Area=
Bretschneider′s formula	Area=.
By Heron’s formula	Area=

Quadrilaterals will typically be of standard shapes having four sides like rectangle, square, trapezoid, and kite. The different types of quadrilaterals are as below:

Area of a Parallelogram

A quadrilateral is stated to be a parallelogram whose opposite sides are parallel and equal to each other. This states that a parallelogram has two pairs of parallel sides with diagonals bisecting each other. In a quadrilateral, the sum of any of the two adjacent angles is 180 degrees.

 Consider the above figure:

Area of the parallelogram=.

Area of a Rectangle

A parallelogram is said to be a rectangle if the measurement of each angle is 90°.

Consider the above figure Here, AB || CD, BC || AD, AB = CD and BC = AD, also ∠A = ∠B = ∠C = ∠D = 90° (each angle is 90 degrees) In a rectangle, the opposite sides are parallel and equal, with diagonals bisecting each other.

Area of a rectangle=.

Area of a Square

A parallelogram is recognized as a square if all sides are equal and the computation of each angle is 90°. The diagonals of a square bisect each other perpendicularly (which implies they bisect at an angle of 90 degrees).

Consider the above figure Here, AB || CD, BC || AD, and AB = CD = BC = AD, along with ∠A = ∠B = ∠C = ∠D = 90° (each angle is 90 degrees).

Consider ‘a’ to be the side length of a square.

Area of the square= (side length squared).

Area of the square=

Area of a Rhombus

A parallelogram is acknowledged to be a rhombus if its all sides are equal with the angle between the diagonal is 90°. All the sides of a rhombus are equal with opposite sides parallel to each other and diagonals bisecting perpendicularly.

If the length of two diagonals of the rhombus is.

Then the area of a rhombus=

Area of a Trapezium

A quadrilateral is termed a trapezium if it has only one pair of parallel opposite sides with the diagonals bisecting each other in the same ratio.

Considering the above diagram;

If ‘h’ represents the height of a trapezium then the; Area of the trapezium is given by:

Area of a Kite

A quadrilateral is termed a kite if the pair of adjacent sides are of the same length with diagonals crossing each other at right angles.

Considering the above diagram:

The area of a Kite is given by=

.

Properties of Quadrilateral

Some of the important properties of a quadrilateral that are needed to understand the topic more clearly are listed below:

• The summation of all the interior angles of a quadrilateral is 360 degrees.
• Every type of quadrilateral holds four vertices and four sides surrounding four angles.
• Generally, a quadrilateral has sides of distinct lengths and angles of different dimensions. Yet quadrilaterals like squares, rectangles, parallelograms, etc possess the same side and angles.

Applications of Area of a Quadrilateral

1. Land Measurement
   Area of quadrilaterals is used to calculate the size of land plots, especially when the shape is irregular, like in fields or residential plots.
2. Architecture and Construction
   Builders use area formulas to estimate the surface area of floors, walls, and ceilings, which are often rectangular or shaped like other quadrilaterals.
3. Interior Design
   Designers calculate the area of carpets, tiles, wallpapers, or floor designs which often take quadrilateral shapes.
4. Agriculture
   Farmers use area calculations to measure how much land is available for planting crops, especially when the fields are not perfectly rectangular.
5. Fencing and Boundary Planning
   Knowing the area helps in determining how much material is needed to fence or cover a quadrilateral-shaped land or garden.
6. Navigation and Mapping
   Mapmakers (cartographers) and navigators use area concepts to calculate distances and regions, especially in the case of plots and blocks on maps.
7. Real Estate and Property Valuation
   Real estate agents and surveyors calculate the area of quadrilateral-shaped properties to estimate their value or price per square unit.

Solved Examples of Area of a Quadrilateral

With the complete information on the area, its formula, how to calculate let us check out some solved examples of area of a quadrilateral:

Example 1: Locate the area of a rhombus whose diagonals are 10 cm and 8 cm respectively.

Solution: The area of the rhombus = square units.

Given: diagonal 1, =10 cm and diagonal 2, = 8 cm.

Hence:

Area of a rhombus =.

Example 2: Find the area of a kite having diagonals as 16 units and 13 units.

Solution: The area of a kite is determined by the formula:

A =

From the given data the diagonals of the provided kite are, = 16 units and = 13 units.

Substituting the values in the formula:

A =

Example 3: Obtain the area of the rectangle having a length equal to 20 cm and a width is 10cm.

Solution: Area of a rectangle is obtained by the formula:

A = l × b

Here the length i.e. l = 20 cm and the breadth i.e. b = 10 cm.

Substituting the values in the formula:

A = l × b = 20 × 10 = 200 .

We hope that the above article on the Area of a Quadrilateral is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

If you are checking Area of a Quadrilateral article, also check related maths articles:
Surface area of a cylinder	Area of Sphere
Surface Area of Cube	Surface Area of a Cone
Surface Area of a Cuboid	Area of Hollow Cylinder

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