Perimeter of Square
Suppose you have a square garden and you want to put a fence around it. How much fencing material do you need? You need to know the total length around the garden. This total length around a shape is called the perimeter.
A square is a special shape where all four sides are equal and all four angles are 90 degrees. Since all sides are equal, finding the perimeter of a square is simple.
The word "perimeter" comes from the Greek words "peri" (meaning around) and "metron" (meaning measure). So perimeter literally means "measure around."
In this topic, you will learn the formula for the perimeter of a square, how to find the side when the perimeter is known, and how to solve word problems on perimeter.
What is Perimeter of Square - Grade 6 Maths (Mensuration)?
Definition: The perimeter of a square is the total length of its boundary. It is the sum of all four sides.
Since a square has four equal sides, the perimeter is simply four times the length of one side.
Key terms:
- Side (a or s): The length of one side of the square. All four sides are equal.
- Perimeter (P): The total distance around the square.
- Unit: Perimeter is measured in linear units (cm, m, km) — NOT in square units.
Important:
- Perimeter is a length, so its unit is cm, m, or km (not cm² or m²).
- Do not confuse perimeter (distance around) with area (space inside).
Perimeter of Square Formula
Formula:
Perimeter of Square = 4 × side = 4a
Where:
- a (or s) = length of one side of the square
- Perimeter is in the same unit as the side
Finding side from perimeter:
Side = Perimeter ÷ 4 = P/4
This formula is used when the perimeter is given and you need to find the side.
Derivation and Proof
How the formula comes from basic addition:
A square has 4 equal sides. Let each side be a cm.
Step 1: Perimeter = sum of all sides
Perimeter = side + side + side + side
Perimeter = a + a + a + a
Step 2: Simplify
Perimeter = 4a
Example: If each side = 7 cm
- Perimeter = 7 + 7 + 7 + 7 = 28 cm
- Or using the formula: P = 4 × 7 = 28 cm
Both methods give the same answer. The formula P = 4a is just a quicker way of adding the four sides.
Reverse formula:
If the perimeter is 36 cm, then:
- 4a = 36
- a = 36 ÷ 4 = 9 cm
Each side is 9 cm.
Types and Properties
Types of perimeter problems:
Type 1: Finding Perimeter (side given)
- Use P = 4a directly.
- Example: Side = 11 cm → P = 4 × 11 = 44 cm
Type 2: Finding Side (perimeter given)
- Use a = P ÷ 4.
- Example: P = 52 m → Side = 52 ÷ 4 = 13 m
Type 3: Unit Conversion
- Convert the side to the required unit before or after calculation.
- Example: Side = 0.5 m → P = 4 × 0.5 = 2 m = 200 cm
Type 4: Word Problems (fencing, wire, ribbon)
- Find the perimeter to determine the length of wire, ribbon, or fencing needed.
- Example: A square field with side 25 m needs fencing. Total fencing = P = 100 m.
Type 5: Cost Problems
- Find perimeter, then multiply by cost per unit length.
- Example: Fencing costs Rs 50 per m and P = 80 m → Total cost = 80 × 50 = Rs 4,000.
Solved Examples
Example 1: Example 1: Basic Perimeter Calculation
Problem: Find the perimeter of a square with side 15 cm.
Solution:
Given: Side (a) = 15 cm
Using the formula:
- P = 4a
- P = 4 × 15
- P = 60 cm
Answer: The perimeter is 60 cm.
Example 2: Example 2: Finding the Side
Problem: The perimeter of a square is 48 m. Find the length of each side.
Solution:
Given: Perimeter = 48 m
Using the formula:
- Side = P ÷ 4
- Side = 48 ÷ 4
- Side = 12 m
Answer: Each side is 12 m.
Example 3: Example 3: Fencing a Garden
Problem: A square garden has a side of 20 m. Find the cost of fencing it at Rs 35 per metre.
Solution:
Step 1: Find the perimeter.
- P = 4 × 20 = 80 m
Step 2: Find the cost.
- Cost = 80 × 35 = Rs 2,800
Answer: The cost of fencing is Rs 2,800.
Example 4: Example 4: Wire Bent into a Square
Problem: A wire 64 cm long is bent to form a square. Find the side of the square.
Solution:
- The wire forms the perimeter of the square.
- Perimeter = 64 cm
- Side = 64 ÷ 4 = 16 cm
Answer: The side of the square is 16 cm.
Example 5: Example 5: Perimeter in Metres and Centimetres
Problem: A square tile has a side of 30 cm. Find the perimeter in metres.
Solution:
- P = 4 × 30 = 120 cm
- Convert to metres: 120 cm = 120 ÷ 100 = 1.2 m
Answer: The perimeter is 1.2 m (or 120 cm).
Example 6: Example 6: Finding Perimeter with Decimal Side
Problem: A square photo frame has a side of 12.5 cm. Find its perimeter.
Solution:
- P = 4 × 12.5
- P = 50 cm
Answer: The perimeter is 50 cm.
Example 7: Example 7: Ribbon Around a Square Box
Problem: Priya ties a ribbon around a square gift box (once around). The box has a side of 18 cm. She also needs 15 cm extra for the bow. What length of ribbon does she need?
Solution:
Step 1: Perimeter = 4 × 18 = 72 cm
Step 2: Add extra for bow = 72 + 15 = 87 cm
Answer: She needs 87 cm of ribbon.
Example 8: Example 8: Comparing Perimeters
Problem: Square A has a side of 9 cm. Square B has a side of 14 cm. How much greater is the perimeter of Square B?
Solution:
- Perimeter of A = 4 × 9 = 36 cm
- Perimeter of B = 4 × 14 = 56 cm
- Difference = 56 − 36 = 20 cm
Answer: Square B's perimeter is 20 cm greater.
Example 9: Example 9: Side Doubled, Perimeter Change
Problem: A square has a side of 6 cm. If the side is doubled, what happens to the perimeter?
Solution:
- Original perimeter = 4 × 6 = 24 cm
- New side = 6 × 2 = 12 cm
- New perimeter = 4 × 12 = 48 cm
- 48 ÷ 24 = 2
Answer: The perimeter also doubles. When the side is doubled, the perimeter is doubled.
Example 10: Example 10: Square Field Path
Problem: Rohan walks around a square park 3 times for his morning exercise. If the side of the park is 200 m, how far does he walk in total?
Solution:
Step 1: Perimeter = 4 × 200 = 800 m
Step 2: Total distance = 3 × 800 = 2,400 m
Convert to km: 2,400 m = 2.4 km
Answer: Rohan walks 2,400 m or 2.4 km.
Real-World Applications
Perimeter of square in daily life:
- Fencing: To put a fence around a square garden or field, you need to know the perimeter to buy the right amount of fencing.
- Framing: Picture frames, photo frames, and mirror frames are often square. The length of frame material needed = perimeter.
- Tying ribbon: Wrapping ribbon around a square box requires knowing the perimeter.
- Sports: Some running tracks and practice areas are square-shaped. Knowing the perimeter helps measure running distance.
- Borders: Adding a decorative border around a square tablecloth or handkerchief uses the perimeter to find the length of lace or trim needed.
- Construction: Laying bricks or tiles along the edge of a square foundation requires knowing the perimeter.
Key Points to Remember
- Perimeter of a square = 4 × side = 4a.
- Side = Perimeter ÷ 4.
- All four sides of a square are equal.
- Perimeter is measured in linear units (cm, m, km), not in square units.
- If the side is doubled, the perimeter is also doubled.
- If the side is tripled, the perimeter is also tripled.
- A wire bent into a square: the length of the wire = perimeter of the square.
- Do not confuse perimeter (boundary length) with area (space inside).
- Always check that your answer is in the correct unit.
- Perimeter is used for fencing, framing, borders, and any problem involving the boundary of a square shape.
Practice Problems
- Find the perimeter of a square with side 22 cm.
- The perimeter of a square is 100 m. Find the side.
- A wire 96 cm long is bent into a square. Find the length of each side.
- A square park has a side of 150 m. Amit jogs around it 4 times. How far does he jog?
- Find the cost of fencing a square plot with side 30 m at Rs 40 per metre.
- The side of a square is 8.5 cm. Find its perimeter.
- Two squares have sides 5 cm and 12 cm. Find the difference in their perimeters.
- The perimeter of a square is 7.2 m. Find the side in centimetres.
Frequently Asked Questions
Q1. What is the perimeter of a square?
The perimeter of a square is the total length of its boundary — the distance all the way around it. Since a square has 4 equal sides, the perimeter = 4 × side.
Q2. What is the formula for perimeter of a square?
Perimeter = 4 × side, written as P = 4a (where a is the side length). To find the side from perimeter, use: side = perimeter ÷ 4.
Q3. What is the difference between perimeter and area of a square?
Perimeter is the distance around the square (P = 4a, in cm or m). Area is the space inside the square (A = a², in cm² or m²). Perimeter is a length; area is a surface measurement.
Q4. What unit is perimeter measured in?
Perimeter is measured in linear units — cm, m, km, etc. It is NOT in square units (cm² or m²). Square units are for area, not perimeter.
Q5. If the side of a square is doubled, what happens to the perimeter?
The perimeter also doubles. If the original side is a, perimeter = 4a. If the side becomes 2a, new perimeter = 4(2a) = 8a, which is twice 4a.
Q6. How do I find the side if I know the perimeter?
Divide the perimeter by 4. If the perimeter is 60 cm, then side = 60 ÷ 4 = 15 cm. This works because P = 4 × side, so side = P ÷ 4.
Q7. Can the perimeter of a square be in decimal?
Yes. If the side is a decimal (like 7.5 cm), the perimeter will also be a decimal: P = 4 × 7.5 = 30 cm. Sometimes the perimeter is a whole number even when the side is a decimal.
Q8. Is a square a special type of rectangle?
Yes. A square is a rectangle where all four sides are equal. So the rectangle perimeter formula P = 2(l + b) also works for a square, giving P = 2(a + a) = 2(2a) = 4a.
Q9. How is perimeter different from circumference?
Perimeter is the boundary length of straight-sided shapes (like squares, rectangles, triangles). Circumference is the boundary length of a circle. Both measure the distance around a shape.
Q10. If a wire is bent into a square, how do I find the side?
The length of the wire = perimeter of the square. So divide the wire length by 4 to get the side. For example, a 40 cm wire forms a square with side = 40 ÷ 4 = 10 cm.
