Parallelogram

A parallelogram is a two-dimensional geometrical shape, whose sides are parallel with each other. It is made up of four sides, where the pair of parallel sides are equal in length. Also, its opposite angles are equal to each other. In geometry, you must have learned about many 2D shapes and sizes such as circle, square, rectangle, rhombus, etc. All of these shapes have a different set of properties. Also, the area and perimeter formulas of these shapes vary with each other, used to solve many problems.

Parallelogram Definition

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length and the opposite angles are equal in measure. 

In the figure above, you can see, ABCD is a parallelogram, where AB//CD and AD//BC. 

Also, AB = CD and AD = BC

And, ∠A = ∠C & ∠B = ∠D

Area of Parallelogram

Area of a parallelogram is the region occupied by it in a two-dimensional plane. Below is the formula to find the parallelogram area:

Area = Base × Height

In the above figure, //gramABCD,  Area is given by;

Area = a b sin A = b a sin B

where a is the slant length of the side of //gramABCD

and b is the base.

Perimeter of Parallelogram

The perimeter of a parallelogram is the total length covered by its boundaries. Hence,

P = AB + BC + CD + AD

P = 2 (a + b)

Properties of Parallelogram

If a quadrilateral has a pair of parallel opposite sides, then it’s a special polygon called Parallelogram. In a parallelogram;

  • The opposite sides are congruent
  • The opposite angles are congruent
  • The consecutive angles are supplementary
  • If anyone of the angles is a right angle, then all the other angles will be right
  • The two diagonals bisect each other
  • The diagonals separate it into congruent

Types of Parallelogram

There are mainly four types of Parallelogram depending on various factors. The factors which distinguish between all of these different types of parallelogram are angles, sides etc.

  1. In a parallelogram, say PQRS 
    • If PQ = QR = RS = SP are the equal sides, then it’s a rhombus. All the properties are the same for rhombus as for parallelogram.
  2. Other two special types of parallelogram are:
  • Rectangle
  • Square

Examples on Parallelogram

Example- Find the area of a parallelogram whose base is 5 cm and height is 8 cm.

Solution- Given, Base = 5 cm and Height = 8 cm.

We know, Area = Base x Height

Area = 5  × 8 

Area =  40 Sq.cm

Example: Find the area of a parallelogram having length of diagonals to be 10 and 22 cm and an intersecting angle to be 65 degrees.

Solution: We know that the diagonals of a parallelogram bisect each other, hence the length of half the diagonal will be 5 and 11 cm.

The angle opposite to the side b comes out to be 180 – 65 = 115°

We use the law of cosines to calculate the base of the parallelogram –

b² =  5² + 11² – 2(11)(5)cos(115°)

b² =  25 + 121 – 110(-.422)

b² = 192.48

b = 13.87 cm.

After finding the base we need to calculate the height of the given parallelogram.

To find the height we have to calculate the value of θ, so we use sine law

5/sin(θ) = b/sin(115)
θ = 19.06

Now we extend the base and draw in the height of the figure and denote it as ‘h’.

The right-angled triangle (marked with red line) has the Hypotenuse to be 22 cm and Perpendicular to be h.

So

sin θ = h/22

h = 7.184 cm

Area = base × height

A = 13.87 × 7.184

A = 99.645 sq.cm