What is the Area of a Parallelogram Overview
What is the Area of a Parallelogram? A parallelogram is a two-dimensional geometric shape that is defined by having two sets of parallel sides. The area of a parallelogram is a measure of the amount of space it occupies in the plane. It is a crucial concept in geometry and is used in various real-world applications, such as in construction, architecture, and cartography. The area of a parallelogram can be calculated using a simple formula that involves multiplying the base and height of the parallelogram.
What is the Area of a Parallelogram?
The area of a parallelogram is measured in square units and is the total number of unit squares that can fit inside it (like cm2, m2, in2, etc.). It is the area that a parallelogram in two dimensions encloses or encompasses. Let's go back and review what a parallelogram is. A parallelogram is a two-dimensional, four-sided shape that has:
- two equal, opposite sides,
- two intersecting and non-equal diagonals, and
- opposite angles that are equal
Other than rectangles and squares, we frequently encounter other geometric forms in our daily lives. The area of a rectangle is comparable to the area of a parallelogram because a few features of a parallelogram and a rectangle are roughly similar.
What is the Area of a Parallelogram: Formula
The area of a Parallelogram may be determined by multiplying its base by its altitude. As seen in the accompanying diagram, a parallelogram's base and altitude are perpendicular to one another. The formula for the area of a parallelogram is expressed as:
A = b x h
Where:
A = area of the parallelogram
b = base of the parallelogram
h = height of the parallelogram
What is the Area of a Parallelogram: Example
The area of a parallelogram can be obtained by using the formula A = b x h.
Take PQRS to be a parallelogram. Let's calculate its area by counting the squares on a grid of paper.
Total number of complete squares = 16
Total number of half squares = 8
Area = 16 + (1/2) × 8 = 16 + 4 = 20 unit2
Also, we observe in the figure that ST ⊥PQ. By counting the squares, we get:
Side, PQ = 5 units
Corresponding height, ST=4 units
Side × height = 5 × 4 = 20 unit2
Thus, the area of the given parallelogram is the base times the altitude.
How to Calculate the Area of a Parallelogram
The area of a parallelogram can be calculated in three different ways. Firstly, by using the base and height of the parallelogram, secondly, by using the length of the parallel sides and an angle between them, if the height is not given. Thirdly, by using the two diagonals of the parallelogram and an intersecting angle between them. Hence, the area of a parallelogram can be derived through multiple methods.
What is the Area of a Parallelogram Using Sides?
Suppose a and b are the set of parallel sides of a parallelogram, and h is its height; then, based on the length of its sides and its 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, the length of base=5 cm and height = 3 cm
As per the formula, Area = 5 × 3 = 15 sq.cm
What is the Area of a 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 adjacent sides of the parallelogram, 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 adjacent 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.
Read more about the Area of Circle.
What is the Area of a 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 summarized formulas of an area of a parallelogram.
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All Formulas to Calculate the 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 the intersection point of the diagonals (∠DOA or ∠DOC)
What is the Area of a 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 the parallelogram.
Area of a 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 d1 and d2, in vector form.
So, we can write;
a + b = d1
b + (-a) = d2
Or
b – a = d2
Thus,
d1 × d2 = (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,
d1 × d2 = a × b + a × b = 2 (a × b)
a × b = 1/2 (d1 × d2)
Hence,
The area of the parallelogram, when diagonals are given in the vector form becomes:
A = 1/2 (d1 × d2)
Where d1 and d2 are vectors of diagonals.
What is the Area of a Parallelogram: Sample Questions
Calculate the area of a solar sheet that is in the shape of a parallelogram, given that, the base measures 20 in, and the altitude measures 8 in.
Solution:
Using the area of parallelogram formula,
Area of the solar cell sheet = B × H = (20) × (8) = 160 in2
Area of solar cell sheet = 160 in2
The area of a playground which is in the shape of a parallelogram is 2500 in2, with one side measuring 250 in. Find the corresponding altitude using the area of the parallelogram formula.
Solution:
Area of the playground = 2500 in2
Side of a playground = 250 in
Corresponding altitude = 2500/250 = 10 in
the corresponding altitude of the playground measures 10 in.
Find the area of the parallelogram with a 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 square.