As the name suggests, a parallelogram is a type of plane figure with two pairs of opposite sides that are parallel and equal. However, a ‘parallelogram’ is slightly different from a rectangle or square. As a result, the ways to find the area of a parallelogram are also different. Here we will precisely deal with the area of a parallelogram and how to find it.
What is the Area of a Parallelogram
The area of a parallelogram is the total space enclosed by its border in a given two-dimension space.
Formulas
With Base and Height
The formula to calculate the area of a parallelogram when base and height are known is given below:
Area of Parallelogram
Derivation with Example
To understand why the above formula is b × h, convert a parallelogram with base (b) and height (h), into a rectangle as shown in the figure below.
Derivation of Area of Parallelogram Formula
After you convert the parallelogram into a rectangle
The base of the parallelogram is equal to the length of the rectangle.
The height of the parallelogram is equal to the width of the rectangle.
Therefore, the formula of area of a parallelogram (A) = b × h is derived, where b = base of a parallelogram, h = height of a parallelogram.
We will analyze the above formula using an example.
Consider a parallelogram ABCD. Using the square grid, we will find its area by counting the squares.
Area of Parallelogram Formula by Square Units
From the above figure:
Total number of complete squares = 24
Total number of half squares = 8
Area = 24 + (1/2) × 8 = 24 + 4 = 28 unit squares
Also, we observe in the figure that AP ⊥ BC. By counting the squares, we get:
Side, BC = 7 units
Corresponding height, AP = 4 units
Side × height = 7 × 4 = 28 unit squares
So, by counting the squares we verified that A = b × h
Thus, the area of the given parallelogram is base times the altitude using height and base
We have learned how to calculate the area of a parallelogram given its base and height.
Let us solve some examples.
Find the area of a parallelogram whose base is 8 cm and height is 11 cm.
Solution:
As we know, Area (A) = b × h, here b = 8 cm and h =11 cm = 8 x 11 cm2 = 88 cm2
Find the area of a parallelogram with a base 12 cm and an altitude 9 cm.
Solution:
As we know, Area (A) = b × h, here b = 12 cm and h =9 cm (since attitude means height) = 12 x 9 cm2 = 108 cm2
Now we will discuss how to calculate the area of a parallelogram when other information is given.
Without Height
When we don’t know the height of the parallelogram, we use trigonometry to find its area.
We will calculate the area of a parallelogram (A) using its adjacent sides and the angle between the sides.
Area of a Parallelogram Without Height
Derivation
∵ We don’t know the height of a parallelogram, we take an imaginary height that is just opposite the angle x.
From trigonometry concept, we know in a ⊥triangle,
The ratio of height to hypotenuse = sin(angle opposite to the height)
∴ Height (h) = a sin x, here a = hypotenuse, b= base, x = angle between a & b.
Hence, we get A = ab sin(x)
Let us solve some examples.
Find the area of a parallelogram whose adjacent sides are 3 cm and 4 cm, respectively. The angle between the two adjacent sides is 60°.
Solution:
As we know, A = ab sin(x), here a = 3 cm, b = 4 cm, x = 60° = 3 × 4 sin(60) = 12 × = 10.392 cm2
What is the area of a parallelogram whose adjacent sides are 6 cm and 8 cm, respectively? The angle between the two adjacent sides is 90°.
Solution:
As we know, A = ab sin(x), here a = 6 cm, b = 8 cm, x = 90° = 6 × 8 sin(90) = 48 × 1 (∵ sin(90) = 1) = 48 cm2
Note: If the angle between the sides of a parallelogram is 90 degrees, it is a rectangle.
Using Diagonals
Here we will learn how to calculate the area of a parallelogram when the 2 diagonals are given along with its angle of intersection.
Area of Parallelogram Using Diagonals
Note: x + y = 180, since diagonals are straight lines
∴ y = 180 – x
Also, sin(x) = sin(180 -x)
So the formula above is justified
Let us solve an example:
Find the area of a parallelogram with diagonals 18 cm and 14 cm. The diagonals intersect at 42°.
Solution:
As we know, (A) = ½ × d1 × d2 sin(x), here d1 = 18, d2 = 14, x = 42° = ½ × 18 × 14 sin(42°) = 84.31 cm2
In Vector Form
There are two ways to calculate the area of a parallelogram in vector form. The formulas are given below:
1) Using Side Vectors
Area of a Parallelogram in Vector Using Side Vectors
Let us solve an example.
Problem: Finding the area of a parallelogram when SIDE VECTORS are known
Find the area of the parallelogram whose sides are represented by a = i + 2j + 3k and b = 4i + 5j + 6k.
Solution:
As we know, A = |a × b|, here a = i + 2j + 3k, b = 4i + 5j + 6k Now, a × b = i(2 × 6 – 3 × 5) – j(1 × 6 – 3 × 4) + k(1 × 5 – 2 × 4) = i(12 – 15) – j(6 – 12) + k(5 – 8) = {-3i + 6j – 3k} |a × b| = √{(-3)2 + 62 + (-3)2} = √54 = 3√6 ≈ 7.35 square units
2) Using Diagonal Vectors
Area of a Parallelogram in Vector Using Diagonal Vectors
Derivation
As we know,
Area of parallelogram (A) in vector is form = |a × b|, here a & b are vectors representing the adjacent sides, and d1 & d2 are the vectors representing the 2 diagonals
a + b = d1…. (1)
b + (-a) = d2
or, b – a = d2…. (2)
now, d1 × d2 = (a + b) × (b – a)
= a × (b – a) + b × (b – a)
= a × b – a × a + b × b – b × a
Since a × a = 0 and b × b = 0
⇒ a × b – 0 + 0 – b × a
Since a × b = – (b × a),
d1 × d2 = a × b – ( -(a × b))
= 2(a × b)
∴ The area of a parallelogram (A) in terms of diagonal when diagonals are in vector forms,
= 1/2 |d1 × d2|
Let us solve an example.
Finding the area of a parallelogram when DIAGONAL VECTORS are known
Find the area of the parallelogram whose diagonals are represented by d1 = 2i- 3j + 4k and d2 = 2i – j + 2k.
