Perimeter of a Parallelogram - Examples, Exercises and Solutions

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 ABCD :

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

or
P=2(a+b) 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 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:

A1 - The perimeter of the parallelogram = P=3+4+3+4=14

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

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

Suggested Topics to Practice in Advance

  1. Area
  2. Parallelogram
  3. The area of a parallelogram: what is it and how is it calculated?

Practice Perimeter of a Parallelogram

Exercise #1

Given the parallelogram:

101010777AAABBBDDDCCC

Calculate the perimeter of the parallelogram.

Video Solution

Step-by-Step Solution

As in a parallelogram each pair of opposite sides are equal and parallel,

It is possible to argue that:

AC=BD=7 AC=BD=7

AB=CD=10 AB=CD=10

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

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

Answer

34

Exercise #2

Given the parallelogram:

666444AAABBBDDDCCC

Calculate the perimeter of the parallelogram.

Video Solution

Step-by-Step Solution

As in a parallelogram every pair of opposite sides are equal:

AB=CD=6,AC=BD=4 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 4+4+6+6=8+12=20

Answer

20

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:

AAABBBDDDCCC8393

Calculate the perimeter of the parallelogram.

Video Solution

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×h=S AB\times h=S

AB×3=39 AB\times3=39

3AB3=393 \frac{3AB}{3}=\frac{39}{3}

AB=13 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=2×13+2×8=26+16=42 2AB+2AC=2\times13+2\times8=26+16=42

Answer

42

Exercise #4

Look at the parallelogram shown below.

AB = 6

AC = X

The perimeter of the parallelogram is 20.

AAABBBDDDCCC6X

Find X.

Video Solution

Step-by-Step Solution

As in a parallelogram each pair of opposite sides are equal:

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

Calculate X according to the given perimeter:

20=6+6+x+x 20=6+6+x+x

20=12+2x 20=12+2x

2012=2x 20-12=2x

8=2x 8=2x

x=4 x=4

Answer

4

Exercise #5

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

AAABBBDDDCCC

Express the perimeter of the parallelogram in terms of X.

Video Solution

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 AB=CD=4AC=4BD

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

x=x=4AC=4BD x=x=4AC=4BD

We divide by 4:

x4=x4=AC=BD \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+x4+x+x4 P=x+\frac{x}{4}+x+\frac{x}{4}

P=2x+x4+x4=212x P=2x+\frac{x}{4}+\frac{x}{4}=2\frac{1}{2}x

Answer

2.5X cm

Exercise #1

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

AAABBBDDDCCC4X32X

Calculate the area of the parallelogram.

Video Solution

Step-by-Step Solution

As in a parallelogram each pair of opposite sides are equal to each other:

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

Now we will find X through the perimeter:60=2x+4x+2x+4x 60=2x+4x+2x+4x

60=12x 60=12x

x=5 x=5

Now we calculate all the sides of the parallelogram:

AB=CD=4×5=20 AB=CD=4\times5=20

AC=BD=2×5=10 AC=BD=2\times5=10

The area of the parallelogram will be equal to:

CD×3=20×3=60 CD\times3=20\times3=60

Answer

60

Exercise #2

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.

AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Let's call CE as X

According to the data

AB=x+2,AD=x3 AB=x+2,AD=x-3

The perimeter of the parallelogram:

2(AB+AD) 2(AB+AD)

38=2(x+2+x3) 38=2(x+2+x-3)

38=2(2x1) 38=2(2x-1)

38=4x2 38=4x-2

38+2=4x 38+2=4x

40=4x 40=4x

x=10 x=10

Now it can be argued:

AD=103=7,CE=10 AD=10-3=7,CE=10

The area of the parallelogram:

CE×AD=10×7=70 CE\times AD=10\times7=70

Answer

70 cm²

Exercise #3

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

AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

The perimeter of the parallelogram is calculated as follows:

SABCD=AB+BC+CD+DA 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 thatAB=DC=2BC AB=DC=2BC

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

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

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

24=6BC 24=6BC

We divide the two sections by 6

24:6=6BC:6 24:6=6BC:6

BC=4 BC=4

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

AB=DC=2×4=8 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:AABCD=AB×EC A_{ABCD}=AB\times EC

We replace the existing data:

24=8×EC 24=8\times EC

We divide the two sections by 824:8=8EC:8 24:8=8EC:8

3=EC 3=EC

Answer

3 cm

Exercise #4

Calculate the perimeter of the parallelogram ABCD.

CD is parallel to AB.

777121212AAABBBCCCDDD

Video Solution

Answer

38

Exercise #5

Given the parallelogram:

444222AAABBBDDDCCC

Calculate the perimeter of the parallelogram.

Video Solution

Answer

12

Exercise #1

Given the parallelogram:

555222AAABBBDDDCCC

Calculate the perimeter of the parallelogram.

Video Solution

Answer

14

Exercise #2

Given the parallelogram:

555444AAABBBDDDCCC

Calculate the perimeter of the parallelogram.

Video Solution

Answer

18

Exercise #3

Given the parallelogram:

888333AAABBBDDDCCC

Calculate the perimeter of the parallelogram.

Video Solution

Answer

22

Exercise #4

Given the parallelogram:

6.56.56.5AAABBBDDDCCC4.5

Calculate the perimeter of the parallelogram.

Video Solution

Answer

22

Exercise #5

Given the parallelogram:

999333AAABBBDDDCCC

Calculate the perimeter of the parallelogram.

Video Solution

Answer

24

Topics learned in later sections

  1. Perimeter