Formula for the Perimeter of a Parallelogram
One of the distinguishing characteristics of a parallelogram is that the opposite sides are equal and parallel to each other.
Here, \( a \) and \( b \) are the sides of a parallelogram.
The perimeter of the parallelogram in the image is given as:
Perimeter, \( P = a+b+a+b \)
\( ~~~~~~~~~~~~~~~~~~~~~= 2 (a + b) \)
Rapid Recall
Perimeter of a parallelogram, \( P = 2(a + b) \) units
Solved Examples
Example 1: The base of a parallelogram is \( 9 \) cm and one side is \( 13 \) cm in length. Find the perimeter of the parallelogram.
Solution:
The measure of the sides of the parallelogram are:
Base, \( b = 9 \) cm
Side length, \( a = 13 \) cm
Perimeter of the parallelogram, \( P = 2(a + b) \) [Formula for the perimeter of a parallelogram]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 2(13 + 9) \) [Substitute \( 13 \) for \( a \) and \( 9 \) for \( b \)]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 2 \times 22 \) cm [Simplify]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 44 \) cm
Hence, the perimeter of the given parallelogram is \( 44 \) cm.
Example 2: A parallelogram was cut exactly at the middle of its longest side to form two parallelograms. What is the length of the smaller parallelogram, provided that the side that was not cut is \( 16 \) cm and the perimeter of the large parallelogram is \( 120 \) cm.
Solution:
Perimeter of the larger parallelogram \( = 120 \)
So, the perimeter of the smaller parallelogram, \( P = \frac{120}{2} = 60 \) cm
Length of one side of the smaller parallelogram, \( a = 16 \) cm
Perimeter of a parallelogram,
\( P = 2(a + b) \) [Formula for the perimeter of a parallelogram]
\( 60 = 2(16 + b) \) [Substitute \( 16 \) for \( a \) and \( 60 \) for \( P \)]
\( \frac{60}{2} = 16 + b \) [Divide by \( 2 \)]
\( 30 = 16 + b \) [Simplify]
\( 30~-~16 = b \)
\( b = 14 \)
Hence, the length of the smaller parallelogram is \( 14 \) cm.
Example 3: Shawn and Macy were cutting a chocolate bar in the shape of a parallelogram. What would be the longer side of the parallelogram if the perimeter of the chocolate bar is \( 16 \) cm and one of its shorter sides is \( 3 \) cm.
Solution:
Perimeter, \( P = 16 \) cm
Smaller side, \( a = 3 \) cm
Perimeter of a parallelogram,
\( P = 2(a + b) \) [Formula for the perimeter of a parallelogram]
\( 16 = 2(3 + b) \) [Substitute \( 3 \) for \( a \) and \( 16 \) for \( P \)]
\( 16 = 6 + 2b \) [Simplify]
\( 16~-~6 = 2b \)
\( 10 = 2b \)
\( b = \frac{10}{2} = 5 \) cm
Therefore, the longer side of the parallelogram is \( 5 \) cm.
Both the perimeter of a rectangle and a parallelogram is twice the sum of their consecutive side lengths.
The perimeter of a shape is the length of the outer boundary. The area on the other hand, is the space covered by a two-dimensional shape.
The perimeter helps to determine the length of the boundary of an enclosed region. If one needs to find the length of the fence that needs to be erected around a given space, then the perimeter should be determined first.