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A parallelogram is a special type of quadrilateral that is formed by parallel lines. In a parallelogram, both pairs of opposite sides are parallel and equal. Like other 2D shapes, there are two types of measurements associated with parallelogram – perimeter and area.

Let’s understand what is area of parallelogram and what are different formulas to find the area of a parallelogram with examples.

## Area of Parallelogram

The area of a parallelogram is the region or space covered by a parallelogram in a 2D plane and it refers to the total number of unit squares that can fit into it. It is measured in square units such as $\text{cm}^2$, $\text{m}^2$, $\text{ft}^2$, $\text{in}^2$, etc. Base (one of the sides) and height (also called altitude) are used to calculate the area of a parallelogram.

## Area of Parallelogram Formula

The area of the parallelogram can be calculated using different formulas using either the sides or the diagonals. Some of the most commonly used formulas are

- Using Side and Height
- Using Diagonals
- Using Sides

### Area of Parallelogram Using Side and Height

Consider a parallelogram $\text{ABCD}$, such that the adjacent sides are $a$ and $b$ and $h$ is the height (perpendicular distance of side $b$ from the opposite vertex), then the area of a parallelogram is given by the formula $\text{Area} = b \times h$.

### Examples on Area of Parallelogram Using Side and Height

**Example 1:** Find the area of the parallelogram with a base of $5$ cm and height of $7$ cm.

Base of parallelogram $b = 5$ cm

Height of parallelogram $h = 7$ cm

Area of a parallelogram = $b \times h = 5 \times 7 = 35 \text{ cm}^2$.

**Example 2:** Find the area of a parallelogram whose breadth is $8$ cm and height is $11$ cm.

Breadth (base) of parallelogram $b = 8$ cm

Height of parallelogram $h = 11$ cm

Area of a parallelogram = $b \times h = 8 \times 11 = 88 \text{ cm}^2$.

**Example 3:** The base of the parallelogram is thrice its height. If the area is $192 \text{ cm}^2$, find the base and height.

Let height of parallelogram = $x$ cm

Therefore base of parallelogram = $3x$ cm

Area of a parallelogram = $\text{Base} \times \text{Height}$

Therefore $3x \times x = 192 => 3x^2 = 192$

$ => x^2 = \frac{192}{3} => x^2 = 64$

Taking the square root of both sides

$x = \sqrt{64} => x = 8$

Base of parallelogram = $3 \times 8 = 24$ cm

And height of parallelogram = $8$ cm

**Example 4:** The area of a parallelogram is $1250$ sq. cm. Its height is twice its base. Find the height and base.

Let base of parallelogram = $x$ cm

Therefore height of parallelogram = $2x$ cm

Area of a parallelogram = $\text{Base} \times \text{Height}$

Therefore $x \times 2x = 1250 => 2x^2 = 1250$

$=> x^2 = \frac{1250}{2} => x^2 = 625$

Taking the square root of both sides

$x = \sqrt{625} => x = 25$

Therefore height of parallelogram = $2 \times 25 = 50$ cm

And base of parallelogram = $25$ cm

**Example 5:** The area of a playground which is in the shape of a parallelogram is $2500 \text{ ft}^2$, with one side measuring $250$ ft. Find the corresponding altitude.

Area of a parallelogram-shaped playground = $2500 \text{ ft}^2$

Length of one side $b = 250$ ft

Let the height of the corresponding altitude = $h$

Area of a parallelogram = $bh$

Therefore $250 \times h = 2500 => h = \frac{2500}{250}$

$=> h = 10$ ft

Therefore altitude (height) = $10$ ft.

### Area of Parallelogram Using Diagonals

Consider a parallelogram $\text{ABCD}$, such that the adjacent sides are $a$ and $b$ and $\text{AC}$ and $\text{BD}$ are the diagonals intersecting at $\text{O}$ making an angle $\alpha$. If the length of the diagonals is $\text{d}_1$ and $\text{d}_2$, then area of a parallelogram is given by the formula $\text{Area} = \frac{1}{2} d_1 d_2 \sin \alpha$.

### Examples Area of Parallelogram Using Diagonals

**Example 1:** Find the area of a parallelogram whose diagonals measure $8$ cm and $6$ cm and the angle between them is $30^{\circ}$.

Length of first diagonal of a parallelogram $\text{d}_1 = 8$ cm

Length of second diagonal of a parallelogram $\text{d}_2 = 6$ cm

Angle between the diagonals $\alpha = 30^{\circ}$

Area of a parallelogram = $\frac{1}{2} \text{d}_1 \text{d}_2 \sin \alpha$

$= \frac{1}{2} \times 8 \times 6 \times \sin 30^{\circ}$

$= 24 \times \frac{1}{2} = 12 \text{cm}^2$

**Example 2:** Area of a parallelogram whose diagonals are $5$ cm and $7$ cm respectively is $\frac{35 \sqrt{3}}{4} \text{ cm}^2$. Find the angle between the diagonals.

Area of a parallelogram = $\frac{1}{2} d_1 d_2 \sin \alpha$

Here $d_1 = 5$ cm, $d_2 = 7$ cm, and Area = $\frac{35 \sqrt{3}}{4} \text{ cm}^2$

Therefore $\frac{1}{2} \times 5 \times 7 \times \sin \alpha = \frac{35 \sqrt{3}}{4} \text{ cm}^2$

$=> \frac{1}{2} \times 35 \times \sin \alpha = \frac{35 \sqrt{3}}{4}$

$=> \frac{1}{2} \times \sin \alpha = \frac{\sqrt{3}}{4}$

$=> \sin \alpha = \frac{\sqrt{3}}{2}$

$=> \alpha = 60^{\circ}$

Thus angle between the diagonals = $60^{\circ}$.

### Area of Parallelogram Using Sides

Consider a parallelogram $\text{ABCD}$, such that the adjacent sides are $a$ and $b$. Further, if $\alpha$ is the angle between the sides $a$ and $b$, then the area of a parallelogram is given by the formula $\text{Area} = a b \sin \alpha$.

### Examples on Area of Parallelogram Using Sides

**Example 1:** Find the area of a parallelogram whose adjacent sides are $10$ cm and $8$ cm and the angle between them is $30^{\circ}$.

Area of a parallelogram = $a b \sin \alpha$

Here $a = 10$ cm, $b = 8$ cm, and $\alpha = 30^{\circ}$

Area of a parallelogram = $10 \times 8 \times \sin 30^{\circ}$

$= 10 \times 8 \times \frac{1}{2} = 40 \text{ cm}^2$.

## Practice Problems

- Find the area of a parallelogram whose adjacent sides are
- $a = 9$ cm and $b = 12$ cm
- $a = 16$ in and $b = 20$ in

- Find the area of a parallelogram whose two diagonals and the angle between them is
- $d_1 = 4$ cm, $d_2 = 6$ cm and $\alpha = 45^{\circ}$
- $d_1 = 10$ cm, $d_2 = 14$ cm and $\alpha = 90^{\circ}$

- Find the area of a parallelogram whose two sides and the angle between the sides is
- $a = 12$ cm, $b = 18$ cm, $\alpha = 90^{\circ}$
- $a = 8$ cm, $b = 14$ cm, $\alpha = 60^{\circ}$

## FAQs

### What is the area of a parallelogram?

The area of a parallelogram is defined as the region enclosed by it in two-dimensional space. It is measured in square units such as $\text{cm}^2$, $\text{m}^2$, $\text{in}^2$, etc.

### What is the area formula for parallelograms?

There are three commonly used formulas to find the area of a parallelogram. These are

a. $b \times h$, where $b$ is the length of one of the sides and $h$ is its perpendicular distance from the opposite vertex

b. $\frac{1}{2} \text{d}_1 \text{d}_2 \sin \alpha$, where $\text{d}_1$, $\text{d}_2$ are the diagonals and $\alpha$ is the angle between them

c. $a b \sin \alpha$, where $a$, $b$ are the adjacent sides and $\alpha$ is the angle between them

### What is the area of a parallelogram when diagonals are given?

The formula used to find the area of a parallelogram when diagonals are given is $\frac{1}{2} \times \text{d}_1 \times $\text{d}_2 \times \sin \alpha$, where $\text{d}_1$ and $\text{d}_2$ are lengths of diagonals of the parallelogram, and $\alpha$ is the angle between them.

### What is the perimeter of a parallelogram?

To find the perimeter of a parallelogram, add all the sides together. The formula used to find the perimeter of a parallelogram is $2 (a + b)$, where $a$, and $b$ are the adjacent sides of a parallelogram.

## Conclusion

A parallelogram is a special type of quadrilateral that is formed by parallel lines. The three most commonly used formulas for finding the area of a parallelogram are

- $b \times h$, where $b$ is length of one of the sides and $h$ is its perpendicular distance from the opposite vertex
- $\frac{1}{2} \text{d}_1 \text{d}_2 \sin \alpha$, where $\text{d}_1$, $\text{d}_2$ are the diagonals and $\alpha$ is the angle between them
- $a b \sin \alpha$, where $a$, $b$ are the adjacent sides and $\alpha$ is the angle between them

## Recommended Reading

- Area of Polygons – Methods, Formulas & Examples
- Area of Trapezium – Formulas, Methods & Examples
- Area of A Kite – Formulas, Methods & Examples
- Area of a Circle – Formula, Derivation & Examples
- Area of Rhombus – Formulas, Methods & Examples
- Area of Square – Definition, Formula & Examples
- Area of a Triangle – Formulas, Methods & Examples
- Area of Rectangle – Definition, Formula & Examples
- Reference Books
- Sample Papers