Table of Contents

Last modified on August 3rd, 2023

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.

The area of a parallelogram is the total space enclosed by its border in a given two-dimension space.

The formula to calculate the area of a parallelogram when base and height are known is given below:

**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.

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**.**

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.

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.

**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 cm^{2}

**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 cm^{2}

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

Here we will learn how to calculate the area of a parallelogram when the 2 diagonals are given along with its angle of intersection.

**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 × d_{1} × d_{2} sin(x), here d_{1} = 18, d_{2} = 14, x = 42°

= ½ × 18 × 14 sin(42°)

= 84.31 cm^{2}

There are two ways to calculate the area of a parallelogram in vector form. The formulas are given below:

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} + 6^{2} + (-3)^{2}}

= √54

= 3√6

≈ 7.35 square units

**Derivation**

As we know,

Area of parallelogram (A) in vector is form = |a × b|, here a & b are vectors representing the adjacent sides, and d_{1} & d_{2} are the vectors representing the 2 diagonals

a + b = d_{1}…. (1)

b + (-a) = d_{2}

or, b – a = d_{2}…. (2)

now, d_{1} × d_{2} = (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),

d_{1} × d_{2} = 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 |d_{1} × d_{2}|

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 d _{1} = 2i- 3j + 4k and d_{2} = 2i – j + 2k.**

Solution:

As we know,

A = 1/2 |(d_{1} × d_{2})|, here d_{1} = 2i – 3j + 4k, d_{2} = 2i – j + 2k

Now, d_{1} × d_{2} = i((-3) × 2 – (-1) × 4) – j(2 × 2 – 2 × 4) + k(2 × (-1) -2 × (-3))

= i(- 6 + 4) – j(4 – 8) + k(-2 + 6)

= -2i + 4j +4k

Now, magnitude of |d_{1} × d_{2}| = √ {(-2)^{2} + 4^{2} + 4^{2}}

= √36

= 6

∴ A = 1/2 |d_{1} × d_{2}|

= 1/2 × 6 = 3 square units

Last modified on August 3rd, 2023