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.
- 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.
- Other two special types of parallelogram are:
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)
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.
sin θ = h/22
h = 7.184 cm
Area = base × height
A = 13.87 × 7.184
A = 99.645 sq.cm
Hope this discussion has made all your doubts clear regarding Parallelograms and their properties. Register with BYJU’S to learn more about quadrilaterals and other maths concepts.