The perimeter of the rectangle can be considered as one of the most important characteristics of the rectangle. The perimeter is defined as the total distance covered by going around the exterior of a rectangle. The perimeter basically gives us the length of a 2-dimensional figure. In the case of a square, in which all its sides have the same length, the perimeter is equal to four times the length of one of its sides.

Here, we will learn about the formula for the perimeter of a rectangle. We will discover the nature of this formula and use it to solve various problems.

## Formula for the perimeter of a rectangle

The perimeter of a rectangle is defined as the sum of the lengths of all the sides of a rectangle. For any polygon, the perimeter formulas are equal to the total distance around the sides of the polygon.

In the case of a rectangle, the opposite sides are equal, which means that the perimeter of a rectangle is equal to twice the length of the base plus twice the length of the height of the rectangle. With this information, we can derive the formula for the perimeter of the rectangle.

Suppose we have a rectangle that has a base of *b* and a height of *a*:

From the definition of the perimeter, we know that the perimeter of a rectangle is equal to ** p = 2(a + b) units**, where,

- “
*a*” is the height of the rectangle - “
*b*” is the base of the rectangle

### Derivation of the perimeter formula

Since the perimeter is equal to the sum of all the sides of the polygon, we know that in the case of the rectangle, the perimeter is:

⇒ *P* = sum of its four sides

⇒ *P* = *a + b + a + b* (opposite sides are equal)

⇒ P = 2(*a + b*)

Therefore, the following is the formula for the perimeter of the rectangle:

Perimeter = 2(Base + Height)p = 2(a+b) |

## Perimeter of a rectangle – Examples with answers

The following examples use the formula for the perimeter of a rectangle to get the answer. Although each example has its respective solution, it is recommended that you try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

Find the perimeter of a rectangle with a base of 12 m and a height of 5 m.

##### Solution

We have the following data:

- Base = 12 m
- Height = 5 m

We have the formula: Perimeter = 2(base + height), so we substitute the values and we have:

$latex p=2(12+5)$

$latex p=2(17)$

$latex p=34$

Therefore, the perimeter of the rectangle is equal to 34 m.

**EXAMPLE 2**

Find the perimeter of a rectangle with a base of 15 m and a height of 8 m.

##### Solution

We have the following information

- Base = 15 m
- Height = 8 m

We use the formula: Perimeter = 2(base + height) with the given values to obtain:

$latex p=2(15+8)$

$latex p=2(23)$

$latex p=46$

Therefore, the perimeter of the rectangle is equal to 46 m.

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**EXAMPLE 3**

A rectangle has a perimeter of 54 cm and its base is 10 cm, what is the length of its height?

##### Solution

We have the following information:

- Perimeter = 54 cm
- Base = 10 cm

We can rearrange the formula: Perimeter = 2(base + height) and plug in the given values to get:

$latex 54=2(10+a)$

$latex 54=20+2a$

$latex 2a=34$

$latex a=17$

Therefore, the length of the height is equal to 17 cm.

**EXAMPLE 4**

Find the perimeter of a rectangle that has a base of 18 m and a height of 12 m.

##### Solution

We have the following data:

- Base = 18 m
- Height = 12 m

We can use the data in the formula: Perimeter = 2(base + height) to obtain:

$latex p=2(18+12)$

$latex p=2(30)$

$latex p=60$

Therefore, the perimeter of the rectangle is equal to 60 m.

**EXAMPLE 5**

Find the base of a rectangle that has a height of 8 and a perimeter of 46.

##### Solution

We have the following information:

- Height = 8 m
- Perimeter = 46 m

We use the perimeter formula: Perimeter = 2(base + height). We substitute the given values and solve for the base:

$latex 46=2(b+8)$

$latex 46=2b+16$

$latex 2b=30$

$latex b=15$

Therefore, the base of the rectangle is equal to 15 m.

## Perimeter of a rectangle – Practice problems

Use what you have learned about the perimeter of a rectangle to solve the following problems. If you need help with this, you can look at the solved examples above.

## See also

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

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