The formula to calculate the perimeter of a parallelogram

You have probably already realized that it is not necessary to calculate all the edge lengths to find the perimeter.

Let's look at the parallelogram $ABCD$:

The equal edges are marked with the letters $a$ and $b$. Let's note the perimeter of the parallelogram:
$P=a+a+b+b=2a+2B=2\left(a+b\right)$

Now let's do it in a clear way.

The formula to calculate the perimeter of a parallelogram is:
$P=2a+2b$

or
$P=2(a+b)$

There is no difference between both formulas, we can use whichever we want.

The perimeter of the parallelogram is equal to the sum of its four edges (or sides). As we know, in a parallelogram there are two pairs of opposite edges of equal length, therefore, knowing the length of two adjacent sides is enough to calculate the perimeter of the figure.

For example, if we observe the parallelogram $ABCD$, given the length of its sides in cm:

As we have mentioned, the perimeter is the sum of the length of its sides. Consequently, we will note:

$P=3+4+3+4=14$

Solution: The perimeter of the parallelogram is $14cm$.

Examples with solutions for Perimeter of a Parallelogram

Exercise #1

Calculate the perimeter of the given parallelogram:

Step-by-Step Solution

As is true for a parallelogram every pair of opposite sides are equal:

$AB=CD=6,AC=BD=4$

The perimeter of the parallelogram is equal to the sum of all sides together:

$4+4+6+6=8+12=20$

20

Exercise #2

Calculate the perimeter of the given parallelogram.

Step-by-Step Solution

As is true for a parallelogram each pair of opposite sides are equal and parallel,

Therefore it is possible to argue that:

$AC=BD=7$

$AB=CD=10$

Now we can calculate the perimeter of the parallelogram by adding together all of its sides:

$10+10+7+7=20+14=34$

34

Exercise #3

Given the parallelogram whose area is equal to 39 cm² and AC=8 cm and the height of the rectangle is 3 cm:

Calculate the perimeter of the parallelogram.

Step-by-Step Solution

The area of a parallelogram is equal to the side multiplied by the height of that side.

First, find the value of AB using the parallelogram area formula:

$AB\times h=S$

$AB\times3=39$

$\frac{3AB}{3}=\frac{39}{3}$

$AB=13$

Since in a parallelogram all pairs of opposite sides are equal and parallel, we can find the perimeter of the parallelogram:

$2AB+2AC=2\times13+2\times8=26+16=42$

42

Exercise #4

The parallelogram ABCD has a perimeter equal to 80 cm.

Calculate X.

Step-by-Step Solution

Since in a parallelogram each pair of opposite sides are equal and parallel:

$BC=AD=2x$

$AB=CD=x$

Now let's substitute the known data into the formula for calculating the perimeter:

$80=2x\times2+2\times x$

$80=4x+2x$

$80=6x$

Let's divide both terms by 6:

$\frac{80}{6}=\frac{6x}{6}$

$\frac{80}{6}=x$

Let's simplify the fraction by 2

$\frac{40}{3}=x$

$x=\frac{40}{3}$

Exercise #5

The longest sides of a parallelogram are X cm long and are four times longer than the shorter sides.

Express the perimeter of the parallelogram in terms of X.

Step-by-Step Solution

In a parallelogram, each pair of opposite sides are equal and parallel: AB = CD and AC = BD.

Given that the length of one side is 4 times greater than the other side equal to X, we know that:

$AB=CD=4AC=4BD$

Now we replace the data in this equation with out own (assuming that AB = CD = X):

$x=x=4AC=4BD$

We divide by 4:

$\frac{x}{4}=\frac{x}{4}=AC=BD$

Now we calculate the perimeter of the parallelogram and express both AC and BD using X:

$P=x+\frac{x}{4}+x+\frac{x}{4}$

$P=2x+\frac{x}{4}+\frac{x}{4}=2\frac{1}{2}x$

2.5X cm

Exercise #6

Look at the parallelogram shown below.

AB = 6

AC = X

The perimeter of the parallelogram is 20.

Find X.

Step-by-Step Solution

As is true for a parallelogram each pair of opposite sides are equal:

$AB=CD=6,AC=BD=x$

Calculate X according to the given perimeter:

$20=6+6+x+x$

$20=12+2x$

$20-12=2x$

$8=2x$

$x=4$

4

Exercise #7

A parallelogram has one side that is 2 times longer than the other. The length of the smaller side is X.

Express the circumference of the parallelogram in terms of X.

Step-by-Step Solution

As is true of a parallelogram each pair of opposite sides are equal to one another

$AB=CD,AC=BD$

Given that AB > AC

Let's call AC by the name X and therefore:

$AB=2AC=2\times x=2x$

Now we know that:

$AB=CD=2x,AC=BD=x$

The perimeter is equal to the sum of all the sides together:

$2x+x+2x+x=6x$

4X+4

Exercise #8

Below is a parallelogram with a perimeter of 60 and a height of 3.

Calculate the area of the parallelogram.

Step-by-Step Solution

As is true for a parallelogram each pair of opposite sides are equal to one other:

$AB=CD=4x,AC=BD=2x$

To begin we will find X through the perimeter:$60=2x+4x+2x+4x$

$60=12x$

$x=5$

Next we will calculate all of the sides of the parallelogram:

$AB=CD=4\times5=20$

$AC=BD=2\times5=10$

Hence the area of the parallelogram will be equal to:

$CD\times3=20\times3=60$

60

Exercise #9

ABCD is a parallelogram whose perimeter is equal to 24 cm.

The side of the parallelogram is two times greater than the adjacent side (AB>AD).

CE is the height of the side AB

The area of the parallelogram is 24 cm².

Find the height of CE

Step-by-Step Solution

The perimeter of the parallelogram is calculated as follows:

$S_{ABCD}=AB+BC+CD+DA$ Since ABCD is a parallelogram, each pair of opposite sides is equal, and therefore, AB=DC and AD=BC

According to the figure that the side of the parallelogram is 2 times larger than the side adjacent to it, it can be argued that$AB=DC=2BC$

We inut the data we know in the formula to calculate the perimeter:

$P_{ABCD}=2BC+BC+2BC+BC$

We replace the given perimeter in the formula and add up all the BC coefficients accordingly:

$24=6BC$

We divide the two sections by 6

$24:6=6BC:6$

$BC=4$

We know that$AB=DC=2BC$We replace the data we obtained (BC=4)

$AB=DC=2\times4=8$

As ABCD is a parallelogram, then all pairs of opposite sides are equal, therefore BC=AD=4

To find EC we use the formula:$A_{ABCD}=AB\times EC$

We replace the existing data:

$24=8\times EC$

We divide the two sections by 8$24:8=8EC:8$

$3=EC$

3 cm

Exercise #10

ABCD is a parallelogram with a perimeter of 38 cm.

AB is twice as long as CE.

AD is three times shorter than CE.

CE is the height of the parallelogram.

Calculate the area of the parallelogram.

Step-by-Step Solution

Let's call CE as X

According to the data

$AB=x+2,AD=x-3$

The perimeter of the parallelogram:

$2(AB+AD)$

$38=2(x+2+x-3)$

$38=2(2x-1)$

$38=4x-2$

$38+2=4x$

$40=4x$

$x=10$

Now it can be argued:

$AD=10-3=7,CE=10$

The area of the parallelogram:

$CE\times AD=10\times7=70$

70 cm²

Exercise #11

Given the parallelogram:

Calculate the perimeter of the parallelogram.

12

Exercise #12

Given the parallelogram:

Calculate the perimeter of the parallelogram.

14

Exercise #13

Calculate the perimeter of the following parallelogram:

36

Exercise #14

Given the parallelogram:

Calculate the perimeter of the parallelogram.

18

Exercise #15

Given the parallelogram:

Calculate the perimeter of the parallelogram.