Geometry is a field of study which it is about shapes and their properties. The shapes are studied on the basis of their dimensions, composition, angles, and other properties. The field deals with the calculation of area, volume, perimeter, and other parameters of a particular shape.

What is a parallelogram?

A parallelogram is a flat or two-dimensional figure with four sides and each pair of sides standing opposite is parallel. Even the opposite interior angles, as well as sides of a parallelogram, are always equal.

Hence, a parallelogram is a solid figure, a two-dimensional quadrilateral with four sides each opposite pair parallel to the other.

Properties of a parallelogram

• The opposite sides of a parallelogram are always equal.
• The opposite angles of a parallelogram are equal.
• If one of the angles in a parallelogram forms a right angle, then the other three also measure to be a right angle.
• The diagonal of a parallelogram bisect each other at the center of the figure.

Parallelogram Formula

A parallelogram as explained above is a two-dimensional structure studied under geometry. The figure has its own properties as well as parameters. Hence, to determine these parameters there are some standard formulas given. 

The combined calculation of the area and perimeter of the parallelogram is known as the parallelogram formula.

Area of Parallelogram

Area is known as the total surface occupied by the figure and the area of a parallelogram will be the surface covered or occupied by it. For, the calculation of the area of a parallelogram three different methods are provided. The area of a parallelogram can be determined by the means of the height of the figure, Sides lengths or the diagonals.

Area of parallelogram by height

The area of a parallelogram by height is derived by the product of base and height.

Mathematically,

Area of parallelogram = b x h

where,

b is the base of the parallelogram

and h is the height of the parallelogram

Area of parallelogram by side length

The area of a parallelogram by side length is derived by the length of its sides and an adjacent angle.

Mathematically,

Area of parallelogram = ab sinθ

where,

a and b are the length of sides 

and θ is the adjacent angle

Area of parallelogram diagonals 

 Diagonals are the two line segments intersecting at the center of the given figure. The area of a parallelogram is determined by the two diagonals and the angle between these diagonals.

Mathematically,

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

where,

d1 and d2 are the diagonals

and x is the angle between the diagonals

Perimeter of parallelogram

Perimeter is the total length or distance of the boundary of the given figure and perimeter of a parallelogram is the total length of the boundary of a parallelogram. The perimeter of a parallelogram is determined by its length and breadth.

Mathematically the formula of the perimeter is written as 

Perimeter of a parallelogram = 2(a+b)

where,

 a and b are the sides of the parallelogram

Sample Problems

Problem 1. Calculate the perimeter of a parallelogram with sides measuring 6cm and 8cm by using the parallelogram formula.

Solution:

Given

The sides of the parallelogram are 6cm and 8cm.

Let a=6cm and b=8cm

We have,

Perimeter=2(a+b)

=>2(6+8)

=>2 x 14

=>28cm

Problem 2. Calculate the perimeter of a parallelogram with sides measuring 5cm and 4cm by using the parallelogram formula.

Solution:

Given

The sides of the parallelogram are 5cm and 4cm.

Let a=5cm and b=4cm.

We have,

Perimeter=2(a+b)

=>2(5+4)

=>2 x 9

=>18cm

Problem 3. Calculate the perimeter of a parallelogram with sides measuring 12cm and 8cm by using the parallelogram formula.

Solution:

Given

The sides of the parallelogram are 12cm and 8cm.

Let a=12cm and b=8cm

We have,

Perimeter=2(a+b)

=>2(12 +8)

=>40cm.

Problem 4. Calculate the area of a parallelogram with a base of 5cm and a height of 8cm by using the parallelogram formula.

Solution:

Given

The base is 5cm

The height is 8cm

We have,

Area = b x h

=>5 x 8

=>40cm²

Problem 5. Calculate the area of a parallelogram with a base of 20cm and a height of 10cm by using the parallelogram formula.

Solution:

Given

The base is 20cm

The height is 10cm

We have,

Area = b x h

=>20 x 10

=>200cm²