About Properties of Parallelogram
What are Parallelograms?
Before discussing the perimeter of a parallelogram, let’s learn a few concepts about the parallelogram.
A parallelogram, as the name itself includes the term ‘parallel’, is a quadrilateral in which the opposite sides are parallel and are of equal length, which implies that the opposite angles are equal. It is a two dimensional shape that is often seen all around us. From tall buildings to tiny erasers, the objects we see in our daily life are examples of parallelograms. The area of a parallelogram can be determined by multiplying its base and height, and is measured in square units.
Area of Parallelogram = Base \( \times \) Height
Other quadrilaterals like squares, rhombuses and rectangles are considered to be special kinds of parallelograms. Then, what makes a parallelogram a parallelogram? It is the properties that set the parallelogram apart from other quadrilaterals.
Properties of Parallelogram
Knowing the properties of a parallelogram helps us find the missing sides and angles, and to find the perimeter. So, the properties of a parallelogram are:
- The diagonals bisect each other
- The opposite sides are congruent
- If there is one right angle, the other angles will also be right angles
- Opposite angles are congruent
Perimeter of Parallelogram
The perimeter of a parallelogram is equal to the sum of the length of all its sides. The perimeter of a parallelogram is measured in units like meters, centimeters, inches, feet, and so on.
Consider a parallelogram with sides a, b, c, and d.
Perimeter of the parallelogram, P = a + b + c + d.
Since the opposite sides in a parallelogram are equal, we can also express the sides as a = c and b = d.
Therefore, the perimeter will be,
P = a + b + a + b
P = 2a + 2b
Hence, Perimeter of a parallelogram = 2 (a + b).
Rapid Recall
Solved Examples
Example 1: Calculate the perimeter of a parallelogram if the length of its sides are 7 cm and 8 cm.
Solution:
The details in the question are, a = 7 cm and b = 8 cm
We know that, the perimeter of a parallelogram is 2 (a + b)
= 2 (7 + 8 ) [Substitute the values in the formula]
= 2 (15)
= 30
Thus, the perimeter of the parallelogram is 30 cm.
Example 2: Meg wants to decorate her doll house and this has a dimension of 6020 sq.cm. by hanging fairy lights all over it. But she is not sure about the length of wire needed for this. Help Meg decorate her doll house by finding the perimeter of the house.
Solution:
As stated, the measurements of Meg’s doll house = 6020 sq.cm.
Consider, a = 60 cm and b = 20 cm
Perimeter of the doll house, P = 2 (a + b)
= 2 (60 + 20) [Substitute the values in the formula]
= 2 (80) [Simplify]
= 160
Hence, the length of wire required to decorate Meg’s doll house is 160 cm.
Example 3: The perimeter of a children’s comic book is 50 cm and the length of one of its sides is 15 cm. Determine the length of the other side of the book.
Solution:
As stated,
The perimeter of the comic book, P = 50 cm
Length of one of the sides of the book, a = 15 cm
Formula for the perimeter of a parallelogram, P = 2 (a + b)
50 = 2 (15 + b) [Substitute the values in the formula]
50 = 2 (15) + 2(b) [Separating the values]
50 = 30 + 2(b)
50 – 30 = 2 (b) [Subtract]
20 = 2 (b) [Simplify]
\( \frac{20}{2} \) = b
10 = b
Therefore, the length of the other side of the comic book is 10 cm.
The shortest distance between the opposite sides of a parallelogram gives the height of a parallelogram.
Yes, the diagonals of a parallelogram always bisect each other.
No, all the sides of a parallelogram are not equal. A parallelogram only has opposite sides that are equal.
Yes, mainly because a rectangle has two sets of parallel sides and opposite angles that are equal.
Some of the parallelograms we might see every day are the design of buildings, roofs, sheets of paper, desks, erasers, and so on.