Parallelogram

Parallelogram Definition

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length & opposite angles are equal in measures.

Properties of Parallelogram

Area of Parallelogram –

Area = base \(\times\) height

A = a\(\times\)b\(\times\) sinA = b \( \times \)a\(\times\)sinB

 Area of Parallelogram

Properties of parallelogram

If a quadrilateral has a pair of parallel opposite sides, then it’s a special polygon called Parallelogram. A parallelogram has six important properties.

The opp sides of a parallelogram are congruent.
The opp angles of a parallelogram are congruent.
The consecutive angles in a parallelogram are supplementary.
If any one of the angles of a parallelogram is right, then all the other angles will be right.
The two diagonals in the parallelogram bisect each other.
The diagonals of the parallelogram separate it into congruent.

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

In a parallelogram 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.
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 = \(5 \times 8 \;cm^{2}\)

Area = \( 40 \;cm^{2}\)

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.

Parallelogram

The angle opposite to the side b comes out to be 180 – 65 = 115\(^{\circ}\)

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.

Parallelogram

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

Parallelogram

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 \(\times\) height

A = 13.87 \(\times\) 7.184

A = 99.645 \(cm ^{2}\)

Hope this discussion has made all your doubts clear regarding this Parallelogram and their properties. Visit BYJU’S to learn more about quadrilaterals and many more.


Practise This Question

There is a pair of parallel lines which intersects with another pair of parallel lines. The quadrilateral formed is trapezium.