Area Formula for Quadrilaterals

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What is an Area of Quadrilateral Mean By?

The area of the quadrilateral is the region enclosed by the four sides of this polygon. The general formula of the area of a quadrilateral is base * height, also written as b*h and the unit of measurement is m2. There are two types of quadrilaterals – regular and irregular. On this page, you will learn how to find the area of any type of quadrilateral, be it a square or a trapezium. Let’s first understand what an area is, so that you can understand the concept of the area of the Quadrilateral.

Topics covered in this article

You will learn the following topics on this page.

1. Definition of area of a quadrilateral

2. Types of quadrilateral

3. How to calculate the area of a quadrilateral?

4. Properties of a quadrilateral

5. Formulas of the area of different types of quadrilateral

6. Solved examples of the area of a quadrilateral

7. Practice Questions 

Definition of Area of Quadrilateral

An area is defined as any region which is included in a particular boundary or figure. If that figure has four sides, the region inside these four sides will be called an area of a quadrilateral. The area of a quadrilateral is measured in square units. The standard unit for measuring an area is mainly square metres, also written as m2.

Area of Different Types of Quadrilateral

As explained in the above paragraph, any polygon having four sides is a quadrilateral, so we have many types of quadrilaterals. Some of them are:

  • Square
  • Rectangle
  • Parallelogram
  • Kite
  • Rhombus
  • Trapezium

How to Calculate the Area of a Quadrilateral?

Step 1: Construct a diagonal PR that will join the opposite vertices of the quadrilateral PQRS.

Step 2: From each of these vertices, draw a perpendicular on the diagonal PR.

Step 3: Area of the quadrilateral PQRS = Area of △PQR + Area of △PRS

Therefore, area of quadrilateral PQRS = ( ½ * PR * PT) + (½ * PR * RU), where T and U are the perpendiculars from the vertices P and R on the diagonal.

What are the Properties of a Quadrilateral?

Let’s learn the basic properties of a quadrilateral.

Every quadrilateral has four vertices, four sides, and four angles.

The sum of all the interior angles of a quadrilateral is always 360 degrees.

The lengths of all the four sides of a quadrilateral may or may not be equal. For example, the sides of a square are equal, but all the four sides of a trapezium are not equal.

Area Formulas of Different Types of Quadrilaterals

As the lengths of the four sides of a quadrilateral may or may not be equal, the formulas to find their areas are also different. Let us learn the formulas to find the area of all the types of quadrilateral in a tabular form.

Area of Quadrilateral Formula

The formula for the area of quadrilateral can be found using different methods such as dividing the quadrilateral into two triangles, or by using Heron’s formula or by using sides of the quadrilateral. Now let us discuss all these methods in detail.

Area of Quadrilateral by Dividing it into Two Triangles

Consider a quadrilateral PQRS, of different (unequal) lengths, let us derive a formula for the area of a quadrilateral.

  • We can view the quadrilateral as a combination of 2 triangles, with the diagonal PR being the common base.
  • hand h2 are the heights of triangles PSR and PQR, respectively.
  • Area of quadrilateral PQRS is equal to the sum of the area of triangle PSR and the area of triangle PQR.
  • Area of triangle PSR = (base * height)/2 = (PR * h1)/2
  • Area of triangle PQR = (base * height)/2 = (PR* h2)/2
  • Thus, area of quadrilateral PQRS is,

Hence, the area of a quadrilateral formula is,

Area of Quadrilateral Using Heron’s Formula

We know that Heron’s formula is used to find the area of a triangle if three sides of the triangle are given. Follow the given procedure to find the area of the quadrilateral.

Step 1: Divide the quadrilateral into two triangles using a diagonal whose diagonal length is known.

Step 2: Now, apply Heron’s formula for each triangle to find the area of a quadrilateral.

[If a, b, c are the sides of a triangle, then Heron’s formula to find the area of a triangle is

Area of triangle = √[s(s-a)(s-b)(s-c)] square units

Where “s” is the semi-perimeter of triangle, which is equal to (a+b+c)/2. ]

Step 3: Now add the area of two triangles to get the area of a quadrilateral.

Area of Quadrilateral Using Sides

If the sides of a quadrilateral (a, b, c, d) are given, and two of its opposite angles (θ1 and θ2) are given, then the area of quadrilateral can be calculated as follows:

Where “s” is the semi-perimeter of the quadrilateral.

(i.e.) s = (a+b+c+d)/2

And,  θ= θ12

Area Formula for All Quadrilaterals

The area formulas for different types of quadrilaterals such as square, rectangle, rhombus, kite, parallelogram and trapezium are given below:

Area of Square = a2 square units.

Where “a” is the side length of a square.

Area of Rectangle = l×b square units

Where “l” and “b” are the length and breadth of a rectangle respectively.

Area of Parallelogram = b×h square units

Where “b” and “h” are the base length and height of the parallelogram.

Area of Rhombus = (½)×d1×d2 square units.

Where “d1” and “d2” are the two diagonals of the rhombus.

Area of kite = (½)pq square units.

Where “p” and “q” are the two diagonals of the kite.

Area of Trapezium = (½)(a+b)h square units.

Where “a” and “b” are the side lengths of parallel sides and “h” is the height of the trapezium.

Let us have a look at the table given below for different types of quadrilaterals and the formulas to find the area of each quadrilateral as mentioned above along with the corresponding shapes.

Area of Quadrilateral with Vertices

In coordinate geometry, the area of the quadrilateral can be calculated using the vertices quadrilateral. 

Let A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4) be the vertices of a quadrilateral ABCD.

Here, we can find the area of the quadrilateral in two ways.

Method 1:

To find the area of the quadrilateral ABCD, we have to choose the vertices A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4) of the quadrilateral ABCD in order (counterclockwise direction) and write them column-wise as shown below. 

Observe the directions given in the dark arrows, add the diagonal products, i.e., x1y2, x2y3, x3y4 and x4y1.

(x1y2 + x2y3 + x3y4 + x4y1)….(i)

Now, consider the dotted arrows and add the diagonal products, i.e., x2y1, x3y2, x4y3 and x1y4.

(x2y1 + x3y2 + x4y3 + x1y4)….(ii)

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

(1/2) ⋅ [(x1y2 + x2y3 + x3y4 + x4y1) – (x2y1 + x3y2 + x4y3 + x1y4)]

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

A =(1/2) ⋅ [(x1y2 + x2y3 + x3y4 + x4y1) – (x2y1 + x3y2 + x4y3 + x1y4)]

Method 2:

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 ABCD = Area of triangle ABD + Area of triangle BCD

Note: 

Area of triangke with vertices P(x1, y1), Q(x2, y2) and R(x3, y3) is given by:

(1/2) |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|

Solved Examples of Area of Quadrilateral

Example 1 : Calculate the area of a quadrilateral with the given measurements:

Diagonal = 50 m, Perpendicular height = 60 m and 20 m.

Solution: Area of quadrilateral = ½ * (a+b) * h

Given : a = 60 m , b = 20 m and h = 50 m

Substituting the values in the formula we get,

A = ½ *(60+20) * 50 

    = ½ * 80 *50

    = 40 * 50

    = 2000 m2

Example 2: A quadrilateral has four equal sides each of length 4cm. Find its area?

Solution: A quadrilateral of all sides equal is known as Square. 

Area of a square – a 2, where a is the side of the square.

A = 4m * 4m = 16m2

Example 3: The length of a quadrilateral (rectangle) is 6 m and the breadth is 5m. What would be the area of this rectangle?

Solution: Area of a rectangle  = l *b

Given : l = 6m and b = 5m

Area of a rectangle = 6m * 5m = 30m2

Example 4: A quadrilateral in the shape of a kite has two diagonals of length 6 m and 8 m. Find the area of this quadrilateral?

Solution: Area of a kite = ½ *1d1 *d2

Given : d1 = 6m and d2  = 8m

A = ½ *  6m * 8m = 24m2

Example 5: A rectangle has sides of length of 5 m and 10 m, Find the area of this rectangle?

Solution: Area of Rectangle = length * Breadth

Given: l = 5m and b = 10m

A = 5m * 10m = 50m2

Now that you know what is the area of a quadrilateral and how to find it, it’s time you should practice the given questions to test your knowledge.

Example 1:

In the given quadrilateral ABCD, the side BD = 15 cm and the heights of the triangles ABD and BCD are 5 cm and 7 cm, respectively. Find the area of the quadrilateral ABCD.

Solution:

Diagonal = BD = 15 cm

Heights, h1= 5 cm & h2 = 7 cm

Sum of the heights of the triangles = h1 + h2 = 5 + 7 = 12 cm

Thus, area of quadrilateral ABCD is:

= (15 * 12)/2 = 90 cm2

Example 2:

Find the area of a rhombus whose diagonals are 7 cm and 6 cm respectively.

Solution:

Given: Diagonal 1, p = 7 cm

Diagonal 2, q = 6 cm.

The area of rhombus = (½)×d1×d2 square units.

A = (½)(7)(6) cm2

A = 7(3) cm2

A = 21 cm2.

Therefore, the area of a rhombus is 21 cm2.

Example: Calculate the area of the quadrilateral formed with the vertices (−3, 2), (5, 4), (7, −6) and (−5, −4).

Solution:

Let A(-3, 2), B(5, 4), C(7, -6) and D(-5, -4) be the vertices of a quadrilateral ABCD.

Thus,

A(-3, 2) = (x1, y1)

B(5, 4) = (x2, y2

C(7, -6) = (x3, y3)

D(-5, -4) = (x4, y4)

We know that, 

Area of quadrilateral ABCD = (1/2) ⋅ [(x1y2 + x2y3 + x3y4 + x4y1) – (x2y1 + x3y2 + x4y3 + x1y4)]

Substituting the values,

= (½). {[-3(4) + 5(-6) + 7(-4) + (-5)2] – {[5(2) + 7(4) + (-5)(-6) + (-3)(-4)]}

= (½).[(-12 – 30 – 28 – 10) – (10 + 28 + 30 + 12)]

= (½) [-80 – 80]

= 160/2 {since area cannot be negative}

= 80

Therefore, the area of the quadrilateral formed with the given vertices is 80 sq. units.

Example 1: Noah measured the sides of the square as 9 m, what would be the area of a square?

Solution:

To find: The area of a square.
Given:
Side of the square = 9 m
Using quadrilateral area formulas,
Area of a Square formula = (side)2
Area of a Square = (9)2
Area = 81 m2

Therefore, the area of a square is 81 m2.

Example 2: If you walk around a trapezoidal park that has one base measuring 200 m and the length of the other base is 100m, with height of the trapezoid shape as 50m, what is the area of that trapezoidal park?

Solution: To find: The area of a trapezoidal park.

One base of a park = 200 m
Second base of a park = 100 m
Height of a park = 50 m
Using quadrilateral area formulas,
Area of a Trapezoid = 1/2 × (Sum of the lengths of parallel sides) × height
= 1/2 × 300 × 50
= 150 × 50
= 7500 m2

Therefore, the area of the trapezoidal park is 7500 m2.

Example 3: The base length of a parallelogram is 7 units and the height is 9 units. Using the quadrilateral area formula of parallelogram find its area.

Solution:

To find the area of a quadrilateral
Given:
Base = 7 units, Height = 9 units
Using quadrilateral area formula of parallelogram
Area of parallelogram formula = Base × height
Area = 7 × 9
Area = 63 units2

Therefore, the area of the quadrilateral is 63 square units.

Example 1: Find the area of the rectangle whose length is 10 in and width is 15 in.

Solution:The length of the rectangle is, l = 10 in.Its breadth is, b = 15 in.Using the formulas of the area of a quadrilateral, the area (A) of the given rectangle is,A = l × b = 10 × 15 = 150 in2.

Answer: The area of the given rectangle = 150 in2.

Example 2: Find the area of a kite whose diagonals are 18 units and 15 units.

Solution:The diagonals of the given kite are, d11 = 18 units and d22 = 15 units.Using the formulas of the area of a quadrilateral, the area (A) of the given kite is,A = (1/2) × d11 × d22 = (1/2) × 18 × 15 = 135 square units.

Answer: The area of the given kite = 135 square units.

Example 3: Find the area of the following quadrilateral. Round your answer to two decimals.

Solution:

The sides of the given quadrilateral are,

a = 15; b = 12; c = 8; and d = 10.

Its semi-perimeter is, s = (a + b + c + d)/2 = (15 + 12 + 8 + 10)/2 = 22.5.

Sum of angles, θ = 100o + 80o = 180o.

The area (A) of the given quadrilateral is found using the Bretschneider′s formula.

Answer: The area of the given quadrilateral = 119.47 square units.

Practice Questions on Quadrilaterals

How do you find the area of a quadrilateral?

Find the area of a parallelogram with base 10 m and height 12m?

Calculate the area of a kite having diagonals of length 13m and 10m?

Which one is not a quadrilateral – triangle, square, rectangle?

Did You Know?

In the word Quadrilateral, Quad means Four and Lateral means sides.

In all types of quadrilaterals, SQUARE is the only regular quadrilateral; the rest all are irregular quadrilaterals.

The sum of all the angles of a quadrilateral is 360 degrees.

Key points on the area of quadrilateral

  1. A Quadrilateral can be both of regular shape or irregular shape.
  2. Some other examples of a Quadrilateral are Kite, rhombus and trapezoid.
  3. The area of a quadrilateral is measured in square units like square metre, square centimetre etc.

FAQs on Area of Quadrilateral

1. What is an area of a quadrilateral where the coordinates of vertices are (1,2), (6,2), (5,3) and (3,4)?

The area of the quadrilateral will be 5.5 square units. To find this, a Quadrilateral ABCD will be drawn with a diagonal division and by adding the area of the two triangles ABC and ACD.

2. What are the Properties of a Quadrilateral?

Let’s learn the basic properties of a quadrilateral.

1.Every quadrilateral has four vertices, four sides, and four angles.

2.The sum of all the interior angles of a quadrilateral is always 360 degrees.

3. The lengths of all the four sides of a quadrilateral may or may not be equal. For example, the sides of a square are equal, but all the four sides of a trapezium are not equal.

What is the area of a quadrilateral?

The area of the quadrilateral is the space occupied by the shape quadrilateral in the two-dimensional space. As we know, a quadrilateral is a 2D figure with four sides. Generally, a quadrilateral is the combined form of a regular or an irregular triangle.

Mention the different types of quadrilateral.

The different types of a quadrilateral are:
Square
Rectangle
Rhombus
Kite
Parallelogram
Trapezium

How to calculate the area of a quadrilateral?

The quadrilateral is the combination of the basic geometric shape called triangles. To calculate the area of a quadrilateral, the area of the individual triangles should be computed, and add the area of the individual triangles.

Mention the applications of quadrilaterals.

Quadrilaterals and its area are mostly used in the field of architecture, agriculture, designing, and navigation to find the actual distance with precision.

How to calculate the area of a quadrilateral if one of its diagonals and perpendiculars from the vertices are given?

If the diagonal and the length of the perpendiculars from the vertices are given, then the area of the quadrilateral is calculated as:
Area of quadrilateral = (½) × diagonal length × sum of the length of the perpendiculars drawn from the remaining two vertices.

What Is the Quadrilateral Area Formula for Parallelogram?

The area of a parallelogram is defined as the amount of space covered by a parallelogram in a two-dimensional plane. A parallelogram is a special kind of quadrilateral. It is a four-sided quadrilateral whose area formula is expressed as, product of its base and height, i.e., A = base × height square units.

What Is the Quadrilateral Area Formula for Rectangle?

The area of a rectangle is the number of unit squares covered within the boundary of the rectangle. It is rectangle is a four-sided quadrilateral with opposite side as equal. Its area formula is expressed as product of its length and breadth, i.e., A = length × breadth square units.

What Is the Quadrilateral Area Formula for Square?

The area of a square is the measure of the space or surface occupied by it. It is a four-sided quadrilateral with all sides equal, whose area formula is equal to the product of the length of its two sides, i.e., A = (side)2 square units.

What Is the Quadrilateral Area Formula for Rhombus?

A rhombus is a four-sided quadrilateral whose area formula is equal to half the product of the lengths of the diagonals. The formula to calculate the area of a rhombus when diagonals are given is expressed as
Area of a rhombus = (1 ⁄ 2) × product of diagonals square units.

What Is the Area of Quadrilateral in Math?

The area of a quadrilateral is the region that is enclosed by it. It is measured in square units such as in2, cm2, m2, etc.

What Is the Area of Quadrilateral Formula?

The area of a quadrilateral can be found by dividing into two triangles using a diagonal. When the diagonal’s length and the heights of the two triangles are known, the area (A) of the quadrilateral is, A = (1/2) × Diagonal × (Sum of heights).

What Is the Area of Quadrilateral Formula Using Sides and Angles?

When 4 sides of the quadrilateral a, b, c, and d and the sum of two of its opposite angles θ are known, then its area is found using the formula

where ‘s’ is the semi-perimeter of the quadrilateral. i.e., s = (a + b + c + d)/2.

How To Find the Area of a Quadrilateral Using Heron’s Formula?

We know that the area of a quadrilateral can be found by dividing it into two triangles using a diagonal. Also, we know that the area of a triangle with 3 sides can be found using Heron’s formula. Using Heron’s formula, the area of the triangle with sides a, b, and c is given by

where ‘s’ is the semi-perimeter of the triangle. i.e., s = (a + b + c)/2. Using this formula, we can find the areas of the two triangles (that are formed by the quadrilateral) and add them to get the area of the quadrilateral.

What Are the Formulas To Find the Areas of Different Types of Quadrilaterals?

There are different formulas used to find the areas of different types of quadrilaterals. They are:

  • The area of a square of side length ‘x’ is x2.
  • The area of a rectangle of dimensions ‘l’ and ‘b’ is l × b.
  • The area of a parallelogram of base ‘b’ and height ‘h’ is b × h.
  • The area of a trapezoid whose parallel sides are ‘a’ and ‘b’ and height (the perpendicular distance between ‘a’ and ‘b’) ‘h’ is (1/2) (a + b) h.

How To Find the Area of a Quadrilateral With Coordinates?

When a quadrilateral’s vertices are given with coordinates, then find the 4 side lengths and the length of a diagonal using the distance formula first. Then divide the quadrilateral into two parts using the diagonal you found, find the area of each triangle using Heron’s formula and then add the areas of two triangles which gives the area of the quadrilateral.

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