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 $14cm$.

The lengths of the sides are shown in cm. Calculate the perimeter of the parallelogram.

We will notice that it is not necessary to calculate the length of each side (or edges). Let's use the formula we just learned to calculate the perimeter of the parallelogram:

$P=2\left(a+b\right)$

Knowing that a and b are the dimensions of the two adjacent sides. Let's place the given numbers and we will obtain: $P=2\left(a+b\right)=2\left(2+6\right)=2\times8=16$

Solution: The perimeter of the parallelogram is $16cm$.

Example 3

Given that the perimeter of the parallelogram is $16cm$. Likewise, we know that the length of one of the sides is $6cm$. How long is the other side?

First, let's draw a parallelogram $ABCD$

Given $P=16$

Let's mark the lengths of the sides with the letters $a$ and $b$. We know that $a=6$ Let's use the formula we just learned: $P=2a+2b$

Let's place the data in the formula and we will get:

$P=2\times6+2\times b=16$

$12+2b=16$

$2b=4$

$b=2$

That is, we have found that the length of the other side is $2cm$.

We can verify our result by doing the following calculation: $a+a+b+b=6+6+2+2=16$

Based on the given data, we are asked to find the perimeter of the parallelogram. As we have already mentioned, opposite sides of a parallelogram are equal, therefore: $AB=CD=12$

$AB=CD=12$

$BC=DA=4$

$P=12\times2+4\times2=32$

The perimeter of the parallelogram is $32cm$.

If you are interested in learning more about the perimeters of geometric shapes, you can enter one of the following articles:

Parallel lines

Parallelogram - Checking the parallelogram

The area of the parallelogram: what is it and how is it calculated?

The perimeter of the rectangle

Rectangles with equivalent area and perimeter

How is the perimeter of a triangle calculated?

How is the perimeter of a trapezoid calculated?

The perimeter of the circumference

Ways to identify parallelograms

Rotational symmetry in parallelograms

On the website ofTutorelayou will find a variety of articles about mathematics.

Perimeter of a Parallelogram Exercises

Exercise 1:

Statement

Given the parallelogram $ABCD$

Given that:

$AB=4$

$AC=x-2$

The perimeter of the parallelogram is equal to $10$

Find $x$

Solution

We use the formula for calculating the perimeter of the parallelogram

$P=2\times AB+2\times AC=10$

We replace the existing data in the formula

$P=2\times4+2\times\left(x-2\right)=10$

We solve accordingly

$P=8+2x-4=10$

$P=4+2x=10$

We move the $4$ to the right section and keep the corresponding sign

$P=2x=10-4$

$P=2x=6$

We divide by: $2$

$P=x=3$

Answer

$3$

Exercise 2:

Statement

Given the parallelogram $ABCD$

Given that:

$AB=6$

$AC=x$

The perimeter of the parallelogram is equal to $20$

Find $x$

Solution

We use the formula for calculating the perimeter of the parallelogram

$P=2\times AB+2\times AC=20$

We replace the existing data in the formula

$P=2\times6+2\times x=20$

We solve accordingly

$P=12+2x=20$

We move the $12$ to the right section and keep the corresponding sign

$P=2x=20-12$

$P=2x=8$

We divide by: $2$

$P=x=4$

Answer

$4$

Exercise 3:

Statement

Given the parallelogram $ABCD$

Given that:

$AB=8$

$AC=x+2$

The perimeter of the parallelogram is equal to $30$

Find $x$

Solution

We use the formula for calculating the perimeter of the parallelogram

$P=2\times AB+2\times AC=30$

We replace the existing data in the formula

$P=2\times8+2\times\left(x+2)=30\right)$

We solve accordingly

$P=16+2x+4=30$

$P=20+2x=30$

We move the $20$ to the right section and keep the corresponding sign

$P=2x=30-20$

$P=2x=10$

We divide by: $2$

$P=x=5$

Answer

$5$

Exercise 4:

Statement

Given the parallelogram $ABCD$

Given that:

$AB=10$

$AC=x$

The perimeter of the parallelogram is equal to $30$

Find $x$

Solution

We use the formula for calculating the perimeter of the parallelogram

$P=2\times AB+2\times AC=30$

We replace the existing data in the formula

$P=2\times10+2\times x=30$

We solve accordingly

$P=20+2x=30$

We move the $20$ to the right section and keep the corresponding sign

$P=2x=30-20$

$P=2x=10$

We divide by: $2$

$P=x=5$

Solution

$5$

Exercise 5:

Statement

Given the parallelogram $ABCD$

Given that:

$AB=7$

$AC=0.5x$

The perimeter of the parallelogram $21$

Find $AC$

Solution

We use the formula for calculating the perimeter of the parallelogram

$P=2\times AB+2\times AC=21$

We replace the existing data in the formula

$P=2\times7+2\times0.5x=21$

We solve accordingly

$P=14+1x=21$

We move the $14$ to the right section and keep the appropriate sign