# Area of Parallelogram

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:

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.

### 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.

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.

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

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

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, ba = d2…. (2)

now, d1 × d2 = (a + b) × (ba)

= a × (ba) + b × (ba)

= a × ba × a + b × bb × 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.

Solution:

As we know,
A = 1/2 |(d1 × d2)|, here d1 = 2i – 3j + 4k, d2 = 2i – j + 2k
Now, d1 × d2 = 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 |d1 × d2| = √ {(-2)2 + 42 + 42}
= √36
= 6
∴ A = 1/2 |d1 × d2|
= 1/2 × 6 = 3 square units