Perimeter is the path that encompasses a shape. The perimeter of a closed geometrical shape (2D shape) is calculated by adding all the sides together. A square is a four-sided polygon, thus its perimeter of the square will be equal to the sum of all its four equal sides.

Let’s learn what is the perimeter of a square and how it is calculated.

## Square – A 2D Plane Figure

A square, in geometry, is a plane 2D shape with four equal sides and four right angles $\left(90^{\circ} \right)$. A square is a special kind of rectangle (an equilateral one) and a special kind of parallelogram (an equilateral and equiangular one).

## What is the Perimeter of a Square?

The perimeter of a square is the total distance covered by its boundaries or sides. Since there are four sides of a square, thus, the perimeter of the square will be the sum of all four sides. Also, the perimeter is a linear measure, therefore, the unit of the perimeter of a square will be in metre, centimetre, inch, feet, etc.

## Perimeter of a Square Formula Using Side

The perimeter of a square is defined as the sum of all the sides of a square. The formula for the perimeter of any polygon is the sum of all its edges (sides). In the case of a square, all four sides are equal and so, if the sides are denoted by $s$, the perimeter of a square is given by the formula $P = 4s$.

### Derivation of Perimeter of Square Formula Using Side

Consider a square $ABCD$, whose length of each side is $s$.

From the figure, you can see lengths are $AB = BC = CD = DA = s$.

The total length of the boundary of a square = $AB + BC + CD + DA = s + s + s + s = 4s$.

### How to Find the Perimeter of a Square Using Side?

You can calculate the perimeter of a square using the following four simple steps.

**Step 1:** Note down the length and width of a square

**Step 2:** Convert the length and width to the same units, if units of length and width are not the same

**Step 3:** Substitute the values of length and width of a square in the formula.

**Step 4:** Simplify the expression in the formula to get the perimeter.

### Examples

**Ex 1:** Find the perimeter of a square whose side measures $8$ cm.

Length of the side of a square $s = 8 cm$

Perimeter of square = $4s = 4 \times 8 = 32 cm$.

Therefore, the perimeter of a square of side length $8 cm$ is $32 cm$.

**Ex 2**: If the perimeter of a square is $56 in$, find the length of the side of the square.

Perimeter of a square $P = 4s$

$=>56 = 4s => s =\frac {56}{4} = 14$.

Therefore, the length of the side of the square perimeter $56 in$ is $14 in$.

**Ex 3: **Mohit wants to fence his square garden of length $8 m$.How much fencing material is needed if want to put fencing three times around the garden?

Length of the square garden $s = 8 m$

Perimeter of the square garden = $4s = 4 \times 8 = 32 m$

Since Mohit wants to put the fence around the garden $3$ times, therefore, the length of fencing material needed is $32 \times 3 = 96 m$.

**Ex 4:** Meenu wants to decorate her square notebook by sticking lace around it. If the total cost of lace is ₹$5.00$ at the rate of ₹$10.00$ per metre, then find the length of the side of the notebook.

Rate of lace = ₹$10.00$ per metre

Cost of lace for Meenu to decorate her notebook is ₹$5.00$

Therefore, the perimeter of a square notebook = $\frac {5}{10} = 0.5 m = 0.5 \times 100 = 50 cm$

$=>4s = 50 => s = \frac{50}{4} = 12.5 cm$.

Therefore, the length of a side of the square notebook = $12.5 cm$.

## Perimeter of a Square Formula Using Diagonal

If you know the length of the diagonal of a square, then also you can find the perimeter of the square. If the length of the diagonal of a square is $d$, then the perimeter of a square is given by the formula $P = 2\sqrt{d}$.

### Derivation of Perimeter of Square Formula Using Diagonal

Consider a square $ABCD$, whose length of each side is $s$.

Here, $AC$ and $BD$ are the diagonals of the square $ABCD$. Let the length of diagonals be $d$.

Using Pythagoras theorem we get, $AC^{2} = AB^{2} + BC^{2} => AC^{2} = s^{2} + s^{2} => AC^{2} = 2s^{2} =>AC = \sqrt{2} s$.

Similarly, $BD = \sqrt{2} s$.

**Note:** The length of the diagonals of a square is equal.

Therefore, $d = \sqrt{2}s => s = \frac {d}{\sqrt{2}}$.

Now, perimeter of square = $4s = 4 \times \frac {d}{\sqrt{2}} = 2\sqrt{2}d$.

### How to Find the Perimeter of a Square Using Length of Diagonal?

You can calculate the perimeter of a square using the following three simple steps.

**Step 1:** Note down the diagonal of a square

**Step 2:** Substitute the values of the diagonal of a square in the formula

**Step 4:** Simplify the expression in the formula to get the perimeter

### Examples

**Ex 1:** Find the perimeter of a square whose diagonal measures $4\sqrt{2} cm$.

Diagonal of a square $d = 4\sqrt{2} cm$.

Perimeter of a square = $2 \sqrt{2}d = 2\sqrt{2} \times 4\sqrt{2} = 2 \times 4 \times \sqrt{2} \times \sqrt{2} = 16 cm$.

**Ex 2:** If the perimeter of a square is $8 in$, then find the diagonal of the square.

Perimeter of a square = $8 in$.

Also, $P = 2\sqrt{2}d => 8 = 2\sqrt{2}d => d = \frac {8}{2\sqrt{2}} = \frac {4}{\sqrt{2}} = 2\sqrt{2} in$.

## Perimeter of Square Formula vs Perimeter of Rectangle Formula

The perimeter of a rectangle formula is different from the formula for the perimeter of a square.

- The perimeter of a rectangle formula is expressed as Perimeter of rectangle = $2\left(l + w \right)$, where $l$ is the length and $w$ is the width. However, the perimeter of a square formula is expressed as, the perimeter of a square = $4 \times s$, where $s$ is the side length.
- The perimeter of a rectangle formula is different from the perimeter of a square because only the opposite sides of a rectangle are equal. However, in the case of a square, all the sides are of equal length.

## Conclusion

The perimeter of any 2D shape is the sum of the lengths of all its sides. Since the length of each side of a square is equal, thus one can find the perimeter of a square which is four times the side. One can also find the perimeter of a square if its diagonal is known using the formula $P = 2\sqrt{d}$.

## Practice Problems

- Find the perimeter of a square whose length of the side is
- $7 cm$
- $12 in$

- Find the side of a square if its perimeter is
- $96 ft$
- $124 m$

- Find the perimeter of a square whose diagonal is of length
- $5 in$
- $3.5 m$

- Find the diagonal of a square whose perimeter is
- $12 m$
- $32 in$

- Manoj wants to fence his square garden of length $12 m$. How much fencing material is needed? What will be the cost of the fencing material available at the rate of ₹$125.00$ per metre?
- A swimmer makes $8$ rounds of square swimming pool swimming the sides of the pool. How much distance does he cover in $10$ rounds?

## Recommended Reading

- What is Length? (With Definition, Unit & Conversion)
- Weight – Definition, Unit & Conversion
- What is Capacity (Definition, Units & Examples)
- What is Time? (With Definition, Facts & Examples)
- What is Temperature? (With Definition & Units)
- Reading A Calendar
- Perimeter of Rectangle – Definition, Formula & Examples

## FAQs

### What is the formula for the perimeter of a square?

The perimeter of a square is the total length around the boundary of a square shape. For a square of length $s$, the perimeter is given by $P = 4s$.

### How to calculate the perimeter of a square?

There are two ways of calculating the perimeter of a square.

One is when the length of the side of a square is known and in this case, the perimeter is given by $P = 4s$, where $s$ is the length of the square.

Another is when the length of the diagonal of a square is known and in this case, the perimeter is given by $P = 2\sqrt{2}d$, where $d$ is the length of the diagonal is $d$.

### How to find the side length of a square when the perimeter is given?

The perimeter of a square is given as $P = 4s$, where $s$ is the length of the side of a square. On rearranging this formula, we get $s = \frac {P}{4}$. Therefore, the side length of a square can be calculated by dividing the given perimeter by $4$.