Area of Parallelogram

What is a Parallelogram?

A parallelogram is a quadrilateral (a flat figure with four sides) in which the opposite sides are parallel and equal to each other in length.

Area of Parallelogram

Area of Parallelogram can be defined as a region which is covered by a parallelogram in a two dimensional plane as a parallelogram is a two dimensional diagram with four sides. As per the definition, the area of the parallelogram is the region the figure covers on a plane and mathematically, it is equal to the product of length and height of the parallelogram.

The sum of a quadrilateral's interior angles is 360 degrees. A parallelogram consists of two pairs of parallel sides that are equal in measurement. Now that it is a two dimensional figure, a parallelogram has an area and a perimeter. Here in this article, we will discuss the area of parallelogram and its formula, how to derive it, and more examples of it in a detailed manner.

A Rectangle, Square, and Rhombus are all examples of a parallelogram. Geometry is all about two dimensional and three dimensional shapes. All these 2D and 3D shapes have a different set of properties with different formulae for area. The primary focus of this article will be on:

• Definition of Area of Parallelogram
• Derivation of Formula for Area of Parallelogram
• Calculation of a Parallelogram's Area in a Vector Form

What is the Area of Parallelogram?

According to the definition, the Area of Parallelogram region enclosed by a parallelogram in a two dimensional plane. As we have discussed above, a parallelogram is a special type of quadrilateral with four sides and each pair of the opposite sides is parallel to each other. Besides, in a parallelogram, the opposite sides are equal in length and the opposite angles are of equal measures. Since the rectangle and the parallelogram have similar properties, the area of the rectangle is equal to that of the parallelogram.

Formula of Area of Parallelogram

To calculate the area of a parallelogram, multiply the perpendicular's base by its height. It should be noted that the parallelogram's base and height are perpendicular to each other, but the parallelogram's lateral side is not. As a result, a dotted line is used to represent the height.

Therefore,

Where -

b = base of the parallelogram

h = height of the parallelogram

In this article, mentioned below is the derivation of the area of a parallelogram.

How to Calculate the Area of Parallelogram?

The area of a parallelogram can be calculated using its base and height. Aside from that, the area of a parallelogram can be calculated if its two diagonals and any of their intersecting angles are known, or if the length of the parallel sides and any of the angles between the sides are known. As a result, there are three methods for calculating the area of a parallelogram:

• When the parallelogram's base and height are given
• When height is not provided
• When given diagonals

Area of Parallelogram using Sides

If a and b are the set of parallel sides of a parallelogram and h is its height, then the formula for its area is given by:

Area = Base (b) x Height (h)

Example 1 - The base of a parallelogram is 5 cm and its height is 3 cm. Find the area of the parallelogram.

Solution 1 - Given,

Length of base = 5 cm

Height = 3 cm

Applying the formula of area of parallelogram -

Area = 5 x 3 = 15 sq.cm

Area of Parallelogram without Height

In case, the height of a parallelogram is not given and not known to us, then we can use Trigonometry concept in order to find the area:

Area = ab sin (x)

Where a and b are the length of parallel sides and x is the angle between the sides of the parallelogram.

Example 2 - If the angle between any two sides of a parallelogram is 90 degrees and if the length of its two parallel sides is 3 cm and 4 cm respectively, then find the area of the parallelogram.

Solution 2 - Let a = 3 cm and b=4 cm

x = 90 degrees

Area = ab sin (x)

A = 3 × 4 sin (90)

A = 12 sin 90

A = 12 × 1 = 12 sq.cm.

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

Area of Parallelogram using Diagonals

We can also calculate the area of any parallelogram using its diagonals' lengths. As we know that there are two diagonals in a parallelogram and these diagonals intersect each other. Assume that the diagonals intersect each other at an angle ‘y', then we can calculate its area from the below mentioned formula:

Area = ½ x d1 x d2 x Sin (y)

The table mentioned below explains the summarized formulae of the area of parallelogram:

Where,

• b = base of the parallelogram (AB)
• h = height of the parallelogram
• a = side of the parallelogram (AD)
• x = any angle between the sides of the parallelogram (∠DAB or ∠ADC)
• d1 = diagonal of the parallelogram (p)
• d2 = diagonal of the parallelogram (q)
• y = any angle between at the intersection point of the diagonals (∠DOA or ∠DOC)

Note: In the above figure,

• DC = AB = b
• AD = BC = a
• ∠DAB = ∠DCB
• ∠ADC = ∠ABC
• O is the intersecting point of the diagonals
• ∠DOA = ∠COB
• ∠DOC = ∠AOB

Area of Parallelogram in Vector Form

If the sides of a parallelogram have been given in the vector form then its area can be calculated using the measurement of its diagonals.

Suppose, vector ‘a' and vector ‘b' are the two sides of a parallelogram, such that the resulting vector is the diagonal of parallelogram.

Area of parallelogram in vector form = Mod of cross-product of vector a and vector b

A = | a × b |

Now, we have to find the area of a parallelogram with respect to diagonals, say d1 and d2,  in vector form.

So, we can write;

a + b = d1

b + (-a) = d2

or

b – a = d2

Thus,

d1 × d2 = (a + b) × (b – a)

= a × (b – a) + b × (b – a)

= a × b – a × a + b × b – b × a

= a × b – 0 + 0 – b × a

= a × b – b × a

Since,

a × b = – b × a

Therefore,

d1 × d2 = a × b + a × b = 2 (a × b)

a × b = 1/2 (d1 × d2)

Hence,

Area of parallelogram when diagonals are given in the vector form, becomes:

A = 1/2 (d1 × d2)

where d1 and d2 are vectors of diagonals.