The perimeter of a parallelogram is defined as the sum of the lengths of all the sides of the parallelogram. The perimeter of the parallelogram is similar to the perimeter of the rectangle. Therefore, by adding the lengths of the parallelogram, we can easily find the perimeter of the parallelogram. It is also possible to find the perimeter of the parallelogram using the length of the base, the length of the height, and an internal angle.

Here, we will learn aout two formulas that we can use to find the perimeter of a parallelogram. In addition, we will use these formulas to solve some problems.

## What is the formula to find the perimeter of a parallelogram?

Let’s use “*a*” and “*b*” to represent the lengths of the sides of a parallelogram. We know that the opposite sides of a parallelogram are parallel and equal in length. Therefore, the formula to find the perimeter of a parallelogram is given by:

$latex p=a+b+a+b$

$latex p=2a+2b$

$latex p=2(a+b)$ |

Therefore, we find the perimeter using the formula $latex p=2(a+b)$.

### Perimeter of a parallelogram with base and height

We can calculate the perimeter of a parallelogram with base and height by using a property of the parallelogram. Suppose that “*b*” is the base and “*h*” is the height of the parallelogram, then, according to the parallelogram property, the opposite sides are parallel and equal and the parallelogram is defined as twice the product of the base and the height multiplied by the cosine of the angle.

Theregore, we have the following formula:

$latex p=2(b+h~\cos(\theta))$ |

where $latex \theta$ is the angle formed between the height and the lateral side of the parallelogram.

## Perimeter of a parallelogram – Examples with answers

The formula for the perimeter of a parallelogram is applied to solve the following examples. Try to solve the examples yourself before looking at the solution to the problem.

**EXAMPLE 1**

If a parallelogram has sides with a length of 8 m and 12 m, what is its perimeter?

##### Solution

We have the following information:

- Side 1, $latex a=8$ m
- Side 2, $latex b=12$ m

Therefore, we use the perimeter formula with these values:

$latex p=2(a+b)$

$latex p=2(8+12)$

$latex p=2(20)$

$latex p=40$

The perimeter is 40 m.

**EXAMPLE 2**

We have a parallelogram with sides of length 15 m and 17 m. What is the perimeter?

##### Solution

We recognize the following values:

- Side 1, $latex a=15$ m
- Side 2, $latex b=17$ m

Therefore, we replace these values in the formula:

$latex p=2(a+b)$

$latex p=2(15+17)$

$latex p=2(32)$

$latex p=64$

The perimeter is 64 m.

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**EXAMPLE 3**

The perimeter of a parallelogram is 90 cm. If one side is 21 cm, how long is the other side?

##### Solution

In this case, we start from the perimeter and want to find the length of the other side. Therefore, we recognize the following:

- Perimeter, $latex p=90$ cm
- Side 1, $latex a=21$ cm

Thus, we use these values and solve for *b*:

$latex p=2(a+b)$

$latex 90=2(21+b)$

$latex 90=42+2b$

$latex 2b=48$

$latex b=24$

The length of the other side is 24 cm.

**EXAMPLE 4**

A perimeter has a height of 10 m and a base with a length of 12 m. If the angle between the height and the lateral side is 60°, what is the perimeter?

##### Solution

We can recognize the following information:

- Height, $latex h=10$ m
- Base, $latex b=12$ m
- Angle, $latex \theta=60$°

Therefore, we use the second formula with these values:

$latex p=2(b+h~\cos(\theta))$

$latex p=2(12+10~\cos(60°))$

$latex p=2(12+10(0.5))$

$latex p=2(12+5)$

$latex p=2(17)$

$latex p=34$

The perimeter is 34 m.

### EXAMPLE 5

A perimeter has a height of 20 m and a base with a length of 15 m. If the angle between the height and the lateral side is 60°, what is the perimeter?

##### Solution

We have the following:

- Height, $latex h=20$ m
- Base, $latex b=15$ m
- Angle, $latex \theta=60$°

Therefore, we substitute these values in the second formula:

$latex p=2(b+h~\cos(\theta))$

$latex p=2(15+20~\cos(60°))$

$latex p=2(15+20(0.5))$

$latex p=2(15+10)$

$latex p=2(25)$

$latex p=50$

The perimeter is 50 m.

## Perimeter of a parallelogram – Practice problems

Put into practice what you have learned about the perimeter of a parallelogram to solve the following problems. Solve the problems and select your answer. Click “Check” to verify that you selected the correct answer.

## See also

Interested in learning more about parallelograms? Take a look at these pages:

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