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 $4$.
