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Area of a Parallelogram

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  • Last Updated : 03 Oct, 2022
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The area of a parallelogram is the space or the region enclosed in a two-dimensional figure within its four sides. A parallelogram in geometry is defined as a special kind of quadrilateral. A parallelogram is a two-dimensional figure which consists of four sides. The opposite sides in a parallelogram are equal and parallel to each other. The area of a parallelogram is simply equal to the number of square units (like m2, cm2, etc) which can fit within the parallelogram.

What is the Area of a Parallelogram?

The total surface occupies by the parallelogram on a two-dimensional plane surface is the area of a parallelogram. The area of a parallelogram can be determined by using its height, length of sides, or diagonals. All these three methods have their own derived formula that is explained below.

What is Parallelogram?

 

Area of Parallelogram Formula

The area of a parallelogram can be determined by multiplying its base by its altitude. As seen in the following diagram, a parallelogram’s base and height are perpendicular to one another. Thus, the following formula can be used to determine a parallelogram’s area:

Area of Parallelogram = Base × Height 

or 

A = b × h

where,

  • b is the base of Parallelogram, and 
  • h is the height of Parallelogram
Area of Parallelogram Formula

 

How to Calculate the Area of Parallelogram?

Generally, the area of a parallelogram can be calculated using its base and height. In addition to that, the area of a parallelogram can also be determined if the lengths of its parallel sides and any angles between them are known, as well as the angles at which its two diagonals intersect. As a result, there are three ways to calculate the parallelogram’s area:

  • When the Base and Height of the parallelogram are known.
  • When the lengths of the sides of the parallelogram are known.
  • When the lengths of the diagonals of the parallelogram are known.

Area of a parallelogram using Base and Height

The area of a parallelogram using the height is given by the product of its base and height.

Mathematically it is written as

Area of parallelogram = b × h

where,

  • b is the base of Parallelogram, and 
  • h is the height of Parallelogram

Example: Find the area of a parallelogram whose base is 12 cm and height is 8 cm. 

Solution:

 Given that, 

 Base (b) = 12 cm

 Height (h) = 8 cm

The formula to calculate the area of a parallelogram is,

A = b × h 

= 12 × 8 

= 96 cm2

Area of a parallelogram using the side lengths

The area of a parallelogram can be calculated by using the length of sides and adjacent angles if the height is not given.

Mathematically it is written as,

Area of parallelogram = ab sin (θ)

where, 

  • a and b are the lengths of parallel sides, and 
  • θ is the angle between the sides.
Area of a parallelogram using the side lengths

 

Example: If the angle between two sides of a parallelogram is 30 degrees and the length of its adjacents sides are 5 cm and 6 cm. Determine the area of the parallelogram.

Solution:

Given that, 

Length of one side (a) = 5 cm,

Length of the other side (b) = 4 cm,

Angle between the two adjacent sides (θ) = 30 degrees

The formula to calculate the area of a parallelogram is,

A = ab sin (θ)

= 5 × 4 × sin (30)

= 10 cm2

Area of parallelogram using diagonals

A parallelogram consists of two diagonals that intersect each other at a certain angle meeting at a particular point. The area of a parallelogram can be calculated by using the length of its diagonals.

The formula for the area of a parallelogram by using the length of diagonals is given by,

Area of parallelogram = 1/2 × d1 × d2 sin (x)

where,

  • d1 and d2 are the lengths of the diagonals and
  • x is the angle between the diagonals.
Area of parallelogram using diagonals

 

Example: Determine the area of the parallelogram, when the angle between two intersecting diagonals of a parallelogram is 90 degrees and the length of its adjacents sides are 2 cm and 6 cm.

Solution:

Given that, 

Length of one diagonal (d1) = 2 cm,

Length of the other diagonal (d2) = 6 cm,

Angle between the two intersecting diagonals (x) = 90 degrees

The formula to calculate the area of a parallelogram is,

A = 1/2 × d1 × d2 sin (x)

= 1/2 × 2 × 6 × sin (90)

= 60 cm2

Area of Parallelogram in Vector Form

The area of a parallelogram can be calculated even when the sides and the diagonals of the parallelogram are given in vector form. Considering a parallelogram PQRS, with adjacent sides \vec a        and \vec b        respectively. And the diagonals are \vec {d_1}        and \vec {d_2}       , as represented below in the diagram:

Area of Parallelogram in Vector Form

 

Now, the area of parallelogram in vector form is given by using the adjacent sides \vec a        and \vec b        as, 

A = |\vec a \times \vec b|

From the above diagram of parallelogram, this can be interpreted as:

\vec a + \vec b = \vec d_1

and 

\vec b + (-\vec a) = \vec d_2

or

\vec b -\vec a = \vec d_2

Now, 

\begin{aligned}\vec d_1 \times \vec d_2 &= (\vec a + \vec b)(\vec b - \vec a)\\&=\vec a \times(\vec b - \vec a)+\vec b\times (\vec b - \vec a)\\&=\vec a \times \vec b - \vec a\times \vec a +\vec b\times \vec b - \vec b \times \vec a)\end{aligned}

But, \vec a \times \vec a = 0, \vec b \times \vec b = 0 and \vec a \times \vec b = - \vec b \times \vec a

Therefore, 

\begin{aligned}\vec d_1 \times \vec d_2 &=\vec a \times \vec b - 0 +0 - \vec b \times \vec a)\\&=\vec a \times \vec b - (-(\vec a \times \vec b))\\&=2(\vec a\times \vec b)\end{aligned}

or

|\vec a + \vec b| = \dfrac{1}{2} |(\vec d_1\times \vec d_2)|

Thus from equation (1), the area of parallelogram in vector form is stated as:

A = \dfrac{1}{2} |(\vec d_1\times \vec d_2)|

Summary of the Area of a Parallelogram

  • Area of a parallelogram using Base and Height:

A = b × h

  • Area of a parallelogram using the side lengths:

A = ab sin (θ)

  • Area of parallelogram using diagonals:

A = 1/2 × d1 × d2 sin (x)

Solved Examples based on the Area of a Parallelogram

Example 1: Find the area of a parallelogram whose base is 10 cm and height is 8 cm.

Solution:

Given:

Base (b) = 10 cm

Height (h) = 8 cm

We have,

A = b × h

= 10 × 8

= 80 cm2

Example 2: Find the area of a parallelogram whose base is 5 cm and height is 4 cm. 

Solution:

Given,

Base (b) = 5cm 

Height (h) = 4cm

Area(A) = b × h 

A = 5 × 4 

= 20 cm2

Example 3: Determine the area of the parallelogram, when the angle between two intersecting diagonals of a parallelogram is 90 degrees and the length of its adjacents sides are 4 cm and 8 cm.

Solution:

Given that, 

Length of one diagonal (d1) = 4 cm,

Length of the other diagonal (d2) = 8 cm,

Angle between the two intersecting diagonals (x) = 90 degrees

The formula to calculate the area of a parallelogram is,

A = 1/2 × d1 × d2 sin (x)

= 1/2 × 4 × 8 × sin (90)

= 16 cm2

Example 4: If the angle between two sides of a parallelogram is 60 degrees and the length of its adjacents sides are 3 cm and 6 cm. Determine the area of the parallelogram.

Solution:

Given that,

Length of one side (a) = 3 cm,

Length of the other side (b) = 6 cm,

Angle between the two adjacent sides (θ) = 60 degrees

The formula to calculate the area of a parallelogram is,

A = ab sin (θ)

= 3 × 6 × sin (60)

= 15.6 cm2

Example 5: Find the area of a parallelogram whose parallel sides are 4 cm and 3 cm and the angle between these sides is 90°.

Solution:

Given,

Let the lengths of the sides by a and b with values 4 cm and 3 cm respectively.

Angle between the sides 90°

Area = ab sinθ

= 4 × 3 sin 90°

= 12 cm2

FAQs on Area of a Parallelogram

Question 1: What is the Perimeter of a Parallelogram?

Answer:

The Perimeter of a parallelogram is defined as the sum of its all four sides, so is given as,

Perimeter of a Parallelogram, P = 2 (a + b)

where a and b are the length of the opposite sides of a parallelogram. 

Question 2: Why is the area of parallelogram base times height?

Answer:

Since a parallelogram is a quadrilateral with two pairs of parallel sides. And perimeter is defined as the distance around a2-D figure. Therefore, the area of a parallelogram is equal to the base multiplied by the height.

Question 3: What are the properties of a parallelogram?

Answer:

The Properties of Parallelogram are:

  • The opposite sides of a parallelogram are equal and parallel to each other.
  • The opposite angles of a parallelogram are equal.
  • The sum of interior angles of a parallelogram is equal to 360°.
  • The adjacent angles of a parallelogram must be supplementary i.e. equal to 180°.

Question 4: Are all sides of a parallelogram equal?

Answer:

No, not all sides of a parallelogram are equal to each other. Only opposite sides of a parallelogram are equal.

Question 5: Are all angles in a parallelogram equal?

Answer:

No, not all angles of a parallelogram are equal to each other. Only opposite angles of a parallelogram are equal.

Question 6: What is the formula for finding the height of a parallelogram?

Answer:

The formula for the height of a parallelogram, when the area and the base of a parallelogram are known then:

h = A/b

where,

  • h is the height, 
  • b is the base and
  • A is the area of the parallelogram

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