Area of Parallelogram
Area of a parallelogram is a region covered by a parallelogram in a two-dimensional plane. In Geometry, a parallelogram is a two-dimensional figure with four sides. It is a special case of the quadrilateral, where opposite sides are equal and parallel. The area of a parallelogram is the space enclosed within its four sides. Area is equal to the product of length and height of the parallelogram.
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. In this article, let us discuss the area of a parallelogram with its formula, derivations, and more solved problems in detail.
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What is the Area of Parallelogram?
The area of a parallelogram is the region bounded by the parallelogram in a given two-dimension space. To recall, a parallelogram is a special type of quadrilateral which has four sides and the pair of opposite sides are parallel. In a parallelogram, the opposite sides are of equal length and opposite angles are of equal measures. Since the rectangle and the parallelogram have similar properties, the area of the rectangle is equal to the area of a parallelogram.
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.
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.
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 × d_{1} × d_{2} 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 × d_{1} × d_{2} 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)
- d_{1} = diagonal of the parallelogram (p)
- d_{2} = diagonal of the parallelogram (q)
- y = any angle between at the intersection point of the diagonals (∠DOA or ∠DOC)
Note: In the above figure,
- DC = AB = b
- AD = BC = a
- ∠DAB = ∠DCB
- ∠ADC = ∠ABC
- O is the intersecting point of the diagonals
- ∠DOA = ∠COB
- ∠DOC = ∠AOB
Area of Parallelogram in Vector Form
If the sides of a parallelogram are given in vector form then the area of the parallelogram can be calculated using its diagonals. Suppose, vector ‘a’ and vector ‘b’ are the two sides of a parallelogram, such that the resulting vector is the diagonal of parallelogram.
Area of parallelogram in vector form = Mod of cross-product of vector a and vector b
A = | a × b|
Now, we have to find the area of a parallelogram with respect to diagonals, say d_{1} and d_{2}, in vector form.
So, we can write;
a + b = d_{1}
b + (-a) = d_{2}
or
b – a = d_{2}
Thus,
d_{1} × d_{2} = (a + b) × (b – a)
= a × (b – a) + b × (b – a)
= a × b – a × a + b × b – b × a
= a × b – 0 + 0 – b × a
= a × b – b × a
Since,
a × b = – b × a
Therefore,
d_{1} × d_{2} = a × b + a × b = 2 (a × b)
a × b = 1/2 (d_{1} × d_{2})
Hence,
Area of parallelogram when diagonals are given in the vector form, becomes:
A = 1/2 (d_{1} × d_{2})
where d1 and d2 are vectors of diagonals.
Example: Find the area of parallelogram whose adjacent sides are given in vectors.
A = 3i + 2j and B = -3i + 1j
Area of parallelogram = |A × B|
= \(\begin{vmatrix} i & j & k\\ 3 & 2 & 0\\ -3 & 1 & 0 \end{vmatrix}\)
= i (0-0) – (0-0) + k(3+6) [Using determinant of 3 x 3 matrix formula]
= 9k
Thus, area of the parallelogram formed by two vectors A and B is equal to 9k sq.unit.
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Solved Examples on Area of Parallelogram
Question 1: Find the area of the parallelogram with the base of 4 cm and 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^{2}
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^{2}
= 88 cm^{2}
Question 3: The base of the parallelogram is thrice its height. If the area is 192 cm^{2}, find the base and height.
Solution:
Let the height of the parallelogram = h cm
then, the base of the parallelogram = 3h cm
Area of the parallelogram = 192 cm^{2}
Area of parallelogram = base × height
Therefore, 192 = 3h × h
⇒ 3 × h^{2} = 192
⇒ h^{2} = 64
⇒ h = 8 cm
Hence, the height of the parallelogram is 8 cm, and breadth is
3 × h
= 3 × 8
Word Problem on Area of Parallelogram
Question: The area of a parallelogram is 500 sq.cm. Its height is twice its base. Find the height and base.
Solution:
Given, area = 500 sq.cm.
Height = Twice of base
h = 2b
By the formula, we know,
Area = b x h
500 = b x 2b
2b^{2} = 500
b^{2} = 250
b = 15.8 cm
Hence, height = 2 x b = 31.6 cm
Practise Questions on Area of a Parallelogram
- Find the area of a parallelogram whose base is 8 cm and height is 4 cm.
- Find the area of a parallelogram with a base equal to 7 inches and height is 9 inches.
- The base of the parallelogram is thrice its height. If the area is 147 sq.units, then what is the value of its base and height?
- A parallelogram has sides equal to 10m and 8m. If the distance between the shortest sides is 5m, then find the distance between the longest sides of the parallelogram. (Hint: First find the area of parallelogram using distance between shortest sides)
Frequently Asked Questions
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^{2}