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Area Of A Parallelogram- Definition, Formula, Examples

Safalta expert Published by: Yashaswi More Updated Tue, 02 Aug 2022 11:36 PM IST

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Check out how to find the area of a parallelogram- definition, formula and examples here at Safalta.com

A parallelogram's area in a two-dimensional plane is the area covered by a parallelogram. In geometry, a parallelogram is a two-dimensional figure with four sides. It's a form of quadrilateral with equal and parallel opposite sides. The area of a parallelogram is the space contained by its four sides. A parallelogram's area is equal to the product of its length and height. Join Safalta School Online and prepare for Board Exams under the guidance of our expert faculty. Our online school aims to help students prepare for Board Exams by ensuring that students have conceptual clarity in all the subjects and are able to score their maximum in the exams.

The sum of the interior angles in a quadrilateral is 360 degrees. A parallelogram has two pairs of parallel sides with equal measures. Since it is a two-dimensional figure, it has an area and perimeter.

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In this article, let us discuss the area of a parallelogram with its formula, derivations, and more solved problems in detail.

Table of Content 

1. What is the Area of Parallelogram?
2. Area of Parallelogram Formula
3. Calculation of the Area of Parallelogram
           i) Area of Parallelogram Using Sides
          ii) Area of Parallelogram Without Height
          iii) Area of Parallelogram Using Diagonals

 


What is the Area of Parallelogram?

The area of a parallelogram is the region in a given two-dimensional space that it circumscribes. A parallelogram is a special type of quadrilateral with four sides, two of which are parallel. A parallelogram has opposite sides of equal length and opposite angles of equal size. The area of a rectangle is equal to the area of a parallelogram because the properties of a rectangle and a parallelogram are identical.

 

Area of Parallelogram Formula

To find the area of the parallelogram, multiply the base of the perpendicular by its height.

It should be noted that the base and the height of the parallelogram are perpendicular to each other, whereas the lateral side of the parallelogram is not perpendicular to the base. Thus, a dotted line is drawn to represent the height.

Therefore,

Area = b × h Square units

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

Let us learn the derivation of area of a parallelogram, in the next section.

 

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How to Calculate the Area of Parallelogram?

The parallelogram area can be calculated, using its base and height. Apart from it, the area of a parallelogram can also be evaluated, if its two diagonals are known along with any of their intersecting angles, or if the length of the parallel sides is known, along with any of the angles between the sides. Hence, there are three method to derive the area of parallelogram:

  • When base and height of parallelogram are given
  • When height is not given
  • When diagonals are given
 

Area of Parallelogram Using Sides

Suppose a and b are the set of parallel sides of a parallelogram and h is the height, then based on the length of sides and height of it, the formula for its area is given by:

Area = Base × Height

A = b × h     [sq.unit]
 

Example: If the base of a parallelogram is equal to 5 cm and the height is 3 cm, then find its area.

Solution: Given, length of base=5 cm and height = 3 cm

As per the formula, Area = 5 × 3 = 15 sq.cm


 

Area of Parallelogram Without Height

If the height of the parallelogram is unknown to us, then we can use the trigonometry concept here to find its area.

Area = ab sin (x)

Where a and b are the length of parallel sides and x is the angle between the sides of the parallelogram.
 

Example: The angle between any two sides of a parallelogram is 90 degrees. If the length of the two parallel sides is 3 cm and 4 cm respectively, then find the area.

Solution: Let a = 3 cm and b=4 cm

x = 90 degrees

Area = ab sin (x)

A = 3 × 4 sin (90)

A = 12 sin 90

A = 12 × 1 = 12 sq.cm.

Note: If the angle between the sides of a parallelogram is 90 degrees, then it is a rectangle.


 

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Area of Parallelogram Using Diagonals

The area of any parallelogram can also be calculated using its diagonal lengths. As we know, there are two diagonals for a parallelogram, which intersects each other. Suppose, the diagonals intersect each other at an angle y, then the area of the parallelogram is given by:

Area = ½ × d1 × d2 sin (y)


 

Check the table below to get summarised formulas of an area of a parallelogram.

All Formulas to Calculate Area of a Parallelogram
Using Base and Height A = b × h
Using Trigonometry A = ab sin (x)
Using Diagonals A = ½ × d1 × d2 sin (y)

Where,

  • b = base of the parallelogram (AB)
  • h = height of the parallelogram
  • a = side of the parallelogram (AD)
  • x = any angle between the sides of the parallelogram (∠DAB or ∠ADC)
  • d1 = diagonal of the parallelogram (p)
  • d2 = diagonal of the parallelogram (q)
  • y = any angle between at the intersection point of the diagonals (∠DOA or ∠DOC)

 

What is a Parallelogram?

A parallelogram is a geometrical figure that has four sides formed by two pairs of parallel lines. In a parallelogram, the opposite sides are equal in length, and opposite angles are equal in measure.

What is the Area of a Parallelogram?

The area of any parallelogram can be calculated using the following formula:

Area = base × height

It should be noted that the base and height of a parallelogram must be perpendicular.

What is the Perimeter of a Parallelogram?

To find the perimeter of a parallelogram, add all the sides together. The following formula gives the perimeter of any parallelogram:

Perimeter = 2 (a + b)

What is the Area of a Parallelogram whose height is 5 cm and base is 4 cm?

The area of a perpendicular with height 5 cm and base 4 cm will be;

A = b × h

Or, A = 4 × 5 = 20 cm square.

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