Perimeter of Square

A square is a fundamental four-sided geometrical shape that has many mathematical and real-life applications. Understanding the area and perimeter of square is essential in mathematics to solve problems and apply them in real-life situations.

1.0What is the Perimeter of a Square?

The perimeter of square definition is characterised by the length covered by the boundaries of the square. A perimeter is a closed geometrical space that is calculated by adding all the sides together. Since a square has 4 equal-length sides, the perimeter of a square can be calculated using a simple formula. To find the perimeter of square, one must multiply the length of one of its sides by 4. Therefore, the formula is: 

$P=4\times s$

Where, 

P is the perimeter of a square; s is the side length of the square. 

Let’s take an example to grasp the concept better. For example, if the side length of a square is 5 cm, then the perimeter will be:

P = 4 x 5 = 20 cm

2.0Derivation of the Perimeter of a Square

To understand the perimeter of square derivation, let’s evaluate its properties first: 

• A square has four equal sides.
• The perimeter of a square is the sum of all four sides.
• Since all the sides are equal, the formula simplifies to 4 times the side length. 

Mathematically, 

P = s + s + s + s; 

P = 4s

This derivation confirms that the perimeter of the square formula is correct and can be used in practical scenarios. 

3.0How to find Perimeter of Square 

Although the most common method of calculating the perimeter of a square is using the side length, there will be multiple instances where other quantities are used instead. These quantities are: 

Perimeter of a Square using Diagonal: 

We know that the diagonal of the square divides the square into two right-angled triangles. This property of a square’s diagonal(d) can be used to calculate the perimeter of a square to find the side length (s) of the square: 

Using Pythagoras: 

$d^2=s^2+s^2$

$d^2=2s^2$

$d=s\sqrt{2}$

$s=\frac{d}{\sqrt{2}}$

Hence, the formula for the perimeter of a square becomes: 

$P=4\frac{d}{\sqrt{2}}$

Perimeter of a Square using Area:

The side length(s) of a square can also be calculated using the formula for the area of a square (A), that is: 

$A=s^2$

$s=\sqrt{A}$

Hence, the formula for the Perimeter of the square (P) becomes: 

$P=4\sqrt{A}$

4.0Difference Between Area and Perimeter of Square

Students need to understand the difference between the area and perimeter of the square formula to have proper fundamental knowledge. Let’s compare the area and perimeter of square in the table below to highlight the differences: 

5.0Practical Applications of Perimeter of a Square

The perimeter of square examples can be found widely in real life, such as:

• Fencing a square garden: Knowing the perimeter of a square area can help you understand the amount of fencing material you will need. 
• Framing a square picture: While framing a square picture, the perimeter is needed to put the border of the frame around the picture.
• Constructing square tiles: While tiling a square area, the perimeter helps understand the tile requirements. 
• Architectural design: Architects use the perimeter of the square formula to understand how much material they need to use. 
• Sports fields: Many sports fields, like small square tennis courts, require the perimeter of a square formula for fencing and boundary marking.

6.0Perimeter of Square Examples

Here are some practical perimeter of square questions that are solved for the students to grasp the concept and apply in their exercises: 

Question 1 : A square has a side length of 7 cm. Find its perimeter.

Solution: By using the perimeter formula, we know P = 4s. If we put the values in their place, then P = 4 x 7. Thereby, s = 28 cm.

Question 2: The perimeter of a square is 40 meters. Find the side length.

Solution: By using the perimeter formula, we know P = 4s. If we put the values in their place, then 40 = 4s; s = 40/4 = 10 cm.

Question 3: A field in the shape of a square has a perimeter of 100 m. How much fencing is required to enclose it?

Solution : Since the perimeter is the total boundary, the fencing required is 100 meters.

Question 4: If the side of a square room is 12 feet, find the total length of the skirting board required along the base.

Solution: By using the perimeter formula, we know P = 4s. If we put the values in their place, then P = 4 x 12 = 48 feet.

Question 5: A square metal sheet has a perimeter of 64 cm. Find its side length and area.

Solution: By using the perimeter formula, we know P = 4s. If we put the values in their place, then 64 = 4s; s = 16 cm.  The formula for the area of a square is A = s²; A = 16² = 256 cm²

7.0Practice Questions for Perimeter of Square

• A city park is designed as a perfect square with a side of 200 m. The local authorities need to install fencing around the park. How much fencing material is required?
• A farmer wants to fence his square farmland with a side length of 500 meters. If fencing costs $10 per meter, what will be the total cost?

8.0Conclusion

The perimeter of a square is a fundamental concept in geometry that is used in mathematics and real-life situations alike. By going through the study guide and understanding the perimeter of square derivations, formulas, and examples, you can easily apply them to solve real problems. Make sure to practice and revise from time to time for a comprehensive grasp of the subject.