# Area of Parallelogram

Area of Parallelogram is the space or the region enclosed by the boundary of the parallelogram in a two-dimensional space. It is simply calculated by multiplying the base of the parallelogram by its height.

## What is Area of Parallelogram?

A Parallelogram’s Area is the total surface covered by the parallelogram in a two-dimensional plane and is expressed in square units. It can be determined by using its height, length of sides, or diagonals. It is measured in the unit square, i.e. m2, cm2.

Formulas to Calculate Area of a Parallelogram

Area of a Parallelogram Using Base and Height

A = b × h

Area of a Parallelogram Using Trigonometry

A = ab sin (x)

Area of a Parallelogram Using Diagonals

A = ½ × d1 × d2 sin (y)

## Area of Parallelogram Formula

Area of a Parallelogram can be determined by multiplying its base by its altitude. Thus, the following formula can be used to determine a parallelogram’s area,

Area of Parallelogram = Base × Height

A = b × h

where,

• b is Base of Parallelogram
• h is Height of Parallelogram

For any parallelogram of base(b) and height(h) whose image is shown below, its area is bh units.

## How to Find Area of Parallelogram?

Area of a parallelogram can be calculated using its base and height. In addition to that, the area of a parallelogram can also be determined if the lengths of its parallel sides and any angles between them are known, as well as the angles at which its two diagonals intersect. As a result,

There are three ways to calculate area of parallelogram:

• When Base and Height of Parallelogram are Known
• When Lengths of Sides of Parallelogram are Known
• When Lengths of Diagonals of Parallelogram are Known

### Parallelogram Area using Base and Height

Area of parallelogram using the height is given by the product of its base and height. Mathematically it is written as

Area of Parallelogram = b × h

Example: Find the area of parallelogram whose base is 12 cm and height is 8 cm.

Given,

•  Base (b) = 12 cm
•  Height (h) = 8 cm

The formula to calculate the area of a parallelogram is,

A = b × h

A = 12 × 8

A = 96 cm2

### Parallelogram Area using Side Lengths

Area of a Parallelogram can be calculated by using the length of sides and adjacent angles if the height is not given. Mathematically it is written as,

Area of Parallelogram = ab sin (θ)

For any parallelogram of sides ‘a’ and ‘b’ and angle between them is ‘θ’ whose image is shown below, its area is ab sin (θ) units.

Example: If the angle between two sides of a parallelogram is 30 degrees and the length of its adjacent sides are 5 cm and 6 cm. Determine the area of parallelogram.

Given,

• Length of One side (a) = 5 cm
• Length of Other side (b) = 4 cm

Angle between two adjacent sides (θ) = 30 degrees

Formula to calculate Area of a Parallelogram is,

A = ab sin (θ)

A = 5 × 4 × sin (30)

A = 10 cm2

### Parallelogram Area using Diagonals

A parallelogram consists of two diagonals that intersect each other at a specific angle meeting at a particular point. The area of a parallelogram can be calculated by using the length of its diagonals.

Formula for the area of parallelogram by using the length of diagonals is given by,

Area of Parallelogram = 1/2 × d1 × d2 sin (x)

For any parallelogram of diagonals ‘d1‘ and ‘d2‘ and angle between them is ‘x’ whose image is shown below, its area is 1/2 × d1 × d2 sin (x) units.

Example: Determine the area of parallelogram, when the angle between two intersecting diagonals of a parallelogram is 90 degrees and the length of its diagonals are 2 cm and 6 cm.

Given,

• Length of One Diagonal (d1) = 2 cm
• Length of Other Diagonal (d2) = 6 cm

Angle between two intersecting diagonals (x) = 90 degrees

Formula to calculate Area of a Parallelogram is,

A = 1/2 × d1 × d2 sin (x)

A = 1/2 × 2 × 6 × sin (90)

A = 6 cm2

## Area of Parallelogram in Vector form

Area of Parallelogram in vector form involves using vectors to express the sides of the parallelogram and then calculating the cross-product of those vectors. The cross-product yields a vector that represents the area of the parallelogram.

A Parallelogram’s area can be calculated even when the sides and the diagonals of the parallelogram are given in vector form. Considering a parallelogram PQRS, with adjacent sides [Tex]\vec a [/Tex]and [Tex]\vec b [/Tex]respectively. And the diagonals are [Tex]\vec {d_1} [/Tex]and [Tex]\vec {d_2}         [/Tex]  .

Now, Area of Parallelogram in vector form is given using adjacent sides [Tex]\vec a [/Tex] and [Tex]\vec b [/Tex] as,

[Tex]A = |\vec a \times \vec b| [/Tex]

[Tex]\vec a + \vec b = \vec d_1 [/Tex]

[Tex]\vec b + (-\vec a) = \vec d_2 [/Tex]

[Tex]\vec b -\vec a = \vec d_2 [/Tex]

Now,

[Tex]\begin{aligned}\vec d_1 \times \vec d_2 &= (\vec a + \vec b)(\vec b – \vec a)\\&=\vec a \times(\vec b – \vec a)+\vec b\times (\vec b – \vec a)\\&=\vec a \times \vec b – \vec a\times \vec a +\vec b\times \vec b – \vec b \times \vec a)\end{aligned} [/Tex]

But, [Tex]\vec a \times \vec a = 0 [/Tex][Tex]\vec b \times \vec b = 0 [/Tex] and [Tex]\vec a \times \vec b = – \vec b \times \vec a [/Tex]

Therefore,

[Tex]\begin{aligned}\vec d_1 \times \vec d_2 &=\vec a \times \vec b – 0 +0 – \vec b \times \vec a)\\&=\vec a \times \vec b – (-(\vec a \times \vec b))\\&=2(\vec a\times \vec b)\end{aligned} [/Tex]

[Tex]|\vec a + \vec b| = \dfrac{1}{2} |(\vec d_1\times \vec d_2)| [/Tex]

Thus from equation (1), the area of the parallelogram in vector form is stated as:

[Tex]A = \dfrac{1}{2} |(\vec d_1\times \vec d_2)| [/Tex]

Example: Find the area of a parallelogram whose adjacent sides are vectors. A = 2i + 5j and B = 7i – j

Area of Parallelogram = |A × B|

A = [Tex]\begin{vmatrix} i& j& k\\ 2& 5& 0\\ 7& -1& 0 \end{vmatrix} [/Tex]

A = i(0-0) – j(0-0) + k(-2-35) = -35k

Area of Prarallelogram is -35k units

## Area of Parallelogram Examples

Various examples related to Area of Parallelogram are,

Example 1: Find area of a parallelogram whose base is 10 cm and height is 8 cm.

Solution:

Given,

• Base (b) = 10 cm
• Height (h) = 8 cm

We have,

A = b × h = 10 × 8 = 80 cm2

Example 2: Find the area of a parallelogram whose base is 5 cm and height is 4 cm.

Solution:

Given,

• Base (b) = 5 cm
• Height (h) = 4 cm

Area(A) = b × h

A = 5 × 4 = 20 cm2

Example 3: Determine the area of the parallelogram, when the angle between two intersecting diagonals of a parallelogram is 90 degrees and the length of its adjacent sides are 4 cm and 8 cm.

Solution:

Given,

• Length of One Diagonal (d1) = 4 cm
• Length of Other Diagonal (d2) = 8 cm

Angle between two intersecting diagonals (x) = 90 degrees

Formula to calculate the area of a parallelogram is,

A = 1/2 × d1 × d2 sin (x)

A = 1/2 × 4 × 8 × sin (90)

A = 16 cm2

Example 4: If the angle between two sides of a parallelogram is 60 degrees and the length of its adjacent sides is 3 cm and 6 cm. Determine the area of the parallelogram.

Solution:

Given,

• Length of One side (a) = 3 cm
• Length of Other side (b) = 6 cm

Angle between two adjacent sides (θ) = 60 degrees

Formula to calculate Area of a Parallelogram is,

A = ab sin (θ)

A = 3 × 6 × sin (60)

A = 15.6 cm2

Example 5: Find the area of a parallelogram whose adjacent sides are 4 cm and 3 cm and the angle between these sides is 90°.

Solution:

Let lengths of sides by a and b with values 4 cm and 3 cm respectively.

Angle between sides 90°

Area = ab sinθ

A = 4 × 3 sin 90°

A = 12 cm2

## Practice Questions on Area of Parallelogram

Some practice questions on Area of parallelogram are,

Q1: Find the area of a parallelogram whose adjacent sides are 12 cm and 14 cm and the angle between these sides is 60°.

Q2: If angle between two sides of a parallelogram is 30 degrees and the length of its adjacent sides is 3 cm and 6 cm. Find its Area.

Q3: If base and height of a parallelogram is 4 cm and 8 cm respectively, find its area.

Q4: What is area of a parallelogram whose breadth is 11 cm and height is 18 cm.

## FAQs on Area of Parallelogram

### What is Area of Parallelogram?

Area of parallelogram is the area inclosed by the  boundary of the parallelogram it can be defined as the 2-D space inside the perimeter of the parallelogram.

### What is Perimeter of Parallelogram?

Perimeter of a parallelogram is defined as the sum of its all four sides, so is given as, P = 2 (a + b)

### What are Properties of Parallelogram?

Properties of Parallelogram are,

• Opposite sides of a parallelogram are equal and parallel to each other.
• Opposite angles of a parallelogram are equal.
• Sum of interior angles of a parallelogram is equal to 360°.
• Adjacent angles of a parallelogram must be supplementary i.e. equal to 180°.

### What is Formula for Finding Height of Parallelogram?

The formula for height of a parallelogram, when area and base of a parallelogram are known then: h = A/b

### How to Calculate Area of Parallelogram?

Area of Paralleogram is calculated using the formula, Area = B × H

### How to Find Area of a Parallelogram without Height?

Area of a parallelogram when height is not given is calculated using the formula, Area = ab sin (θ)

### How to Find Area of Parallelogram when Diagonals are Given?

Area of a parallelogram when diagonals are given is calculated using the formula, Area = 1/2 × d1 × d2 sin (x)

### How to Find Area of Parallelogram in Coordinate Geometry?

If the coordinate of the vertices of the paralleogram A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4) are given then it can be divided into two triangles, triangle ABC and triangle BCD and its area can be calculated using area of triangle with given coordinate.

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