Area of a Parallelogram

Area of a Parallelogram

A parallelogram is a 2-dimensional 4-sided plane figure whose opposite sides are parallel and equal in length. The area of a parallelogram is the space occupied by the boundary of the parallelogram. A rectangle, a square and a rhombus follow the properties of a parallelogram, therefore, these are a parallelogram.

The area of a parallelogram is defined as the number of unit squares covered by the parallelogram is called the area of a parallelogram. If a parallelogram covers 12 unit squares, then its area will be 12 unit square. 

The unit of area of a parallelogram is square units, i.e., sq. cm, sq. m, sq. mm, sq. km, etc.

Area of a Parallelogram Formula

Area of a parallelogram = b × h

Where b is the base and h is the perpendicular height of the parallelogram.

Derivation of Area of a Parallelogram Formula

Let us draw a rectangle ABCD and mark a point E on AB and F on CD as shown in the given diagram.

 Now, join EF and divide the rectangle into two trapeziums. Name them as T₁ and T₂.

 

Then, cut these two trapeziums and place them to form a parallelogram as shown in the following figure.

Now, compare the area of the rectangle and the parallelogram formed.

Let us deduce the formula to find the area of the parallelogram formed in the above activity.

From the above activity, we observe that

Area of the parallelogram formed = Area of the rectangle

Now, Area of the rectangle = Length × breadth

Length of the rectangle = Base of the parallelogram (b)

Breadth of the rectangle = Height of the parallelogram (h)

Therefore, the area of the parallelogram = Area of the rectangle = length × breadth = b × h = bh sq. units

Hence, Area of the parallelogram = b × h

Observation: Area of a parallelogram is bh sq. units, where b is the length of the base and h is the height of the parallelogram.

Area of a Parallelogram When its Two Sides are Given

We know that the area of a parallelogram formula is bh, i.e., base × perpendicular height.

If the length of the two adjacent sides of a parallelogram and angle between them are given and perpendicular height is not given, then we can find the perpendicular height of the parallelogram using the concept of trigonometry.

Suppose the length and the breadth of a parallelogram are a and b, respectively.

Then, base = a and perpendicular height = b sin θ

In this case, area of a parallelogram = ab sin θ

Where a is the length, b is the breadth and θ is the angle between them. 

How to Calculate Area of a Parallelogram?

To calculate the area of a parallelogram, we can follow these steps:

Step 1: Write the length of the base as b. If the length of the base is 5 cm, write b = 5 cm.

Step 2: Write the perpendicular distance between the base and its parallel opposite side as h. If the perpendicular distance between two parallel opposite sides is 4.5 cm, write h = 4.5 cm.

Step 3: Now, apply the formula, area of a parallelogram = bh = 5 × 4.5 = 22.5 sq. cm

Step 4: Multiply the value of b and h to find the area of a parallelogram.

Step 5: If the perpendicular height of the parallelogram is not given and the angle between two adjacent sides are given, then apply the formula,

area of a parallelogram = ab sin θ

Area of a Parallelogram Examples

Example 1: Find the area of a parallelogram whose base is 9 cm and height is 6 cm.

Solution: Here, b = 9 cm and h = 6 cm

Area of a parallelogram = b × h

                                           = 9 × 6 = 54 sq. cm

 

Example 2: Find the height of a parallelogram whose base is 12 cm and area is 120 sq. cm.

Solution: Here, b = 12 cm and area = 120 sq. cm

Height (h) = ?

Area of a parallelogram = b × h

                                   120 = 12 × h

                                   h = 120/12 = 10 cm

 

Example 3: The two sides AB and BC of a parallelogram are 24 cm and 16 cm. The height corresponding to AB is 10 cm. Find the area of the parallelogram and the height corresponding to BC.

 

Solution: Here, b = 24 cm and h = 10 cm

Area of parallelogram = b × h = 24 × 10 = 240 sq. cm

Let x be the height corresponding to the side BC.

Then, the area of the parallelogram = 16 × x

But, 16x = 240

So, x = = 240/16 = 15 cm

Therefore, the height corresponding to side BC is 15 cm.

Example 4: The length and the breadth of a parallelogram are 8.4 cm and 6 cm, respectively.

If the angle between them is 30 degree, find the area of the parallelogram. 

Solution: Here, b = 8.4 cm and h = 6 sin θ = 6 sin θ = 6 × 1/2 = 3 cm

Thus, area of a parallelogram = b h = 8.4 × 3 = 25.2 sq. cm

If we take the adjacent sides as a and b, then a = 8.4 cm and b = 6 cm

Area of a parallelogram = ab sin θ = 8.4 × 6 × sin 30 

                                                        = 8.4 × 6 × 1/2

                                                        = 25.2 sq. cm

To find the area of a trapezium, a rhombus, a rectangle, a square and a circle, click the following links:

Area of a Trapezium

Area of a Rhombus

Area of a Rectangle

Area of a Square

Area of a Circle