Area of a Parallelogram: A Parallelogram is a two-dimensional geometric figure. The opposite sides of a parallelogram are parallel to each other. How to find an area of a parallelogram must be known to all the students.
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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?
A parallelogram's area is the region it encircles in a given two-dimensional space.
A parallelogram is a type of quadrilateral with four parallel sides.
A parallelogram has opposite sides of equal length and opposite angles of equal size.
The area of a rectangle equals the area of a parallelogram since the qualities of a rectangle and a parallelogram are identical.
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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 the 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 methods to derive the area of a parallelogram:
- When the base and height of the 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, 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 lengths 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 intersect 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 × hosting |
| g 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 the intersection point of the diagonals (∠DOA or ∠DOC)
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Solved Examples of Parallelogram
Question 1: Find the area of the parallelogram with a base of 4 cm and a height of 5 cm.
Solution:
Given:
Base, b = 4 cm
h = 5 cm
We know that,
Area of Parallelogram = b×h Square units
= 4 × 5 = 20 sq.cm
Therefore, the area of a parallelogram = 20 cm square.
Question 2: Find the area of a parallelogram whose breadth is 8 cm and height is 11 cm.
Solution:
Given,
b = 8 cm
h = 11 cm
Area of a parallelogram
= b × h
= 8 × 11 cm square
= 88 cm square.
What is a Parallelogram?
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 a height 5 cm and a base 4 cm will be;
A = b × h
Or, A = 4 × 5 = 20 cm square.
