Area of Square
Suppose you have a square tile that is 1 metre on each side. How much floor space does it cover? It covers exactly 1 square metre. Now, if you place 9 such tiles in a 3 × 3 arrangement, the total floor space covered is 9 square metres. This is the area of a square with side 3 m.
The area of a shape tells you how much surface it covers. While perimeter measures the boundary (like the length of a fence), area measures the region inside (like the amount of paint needed to fill it).
Area is measured in square units — square centimetres (cm²), square metres (m²), or square kilometres (km²). A square unit is literally a square with sides of 1 unit each. When you find the area of a shape, you are counting how many such unit squares fit inside it.
Think about tiling a bathroom floor. If the floor is a 4 m by 4 m square, and each tile is a 1 m by 1 m square, you need 4 × 4 = 16 tiles. The area of the floor is 16 m². This simple idea of counting squares is the basis of the area formula.
In this chapter, you will learn the formula for the area of a square, how to use it in different situations, how to find the side of a square when its area is given, and how area changes when sides change. This topic is part of the Mensuration chapter in Grade 6 Maths (NCERT/CBSE).
What is Area of Square - Grade 6 Maths (Mensuration)?
Definition: The area of a square is the amount of space enclosed within its four sides. It is measured in square units (sq. cm, sq. m, sq. km, etc.).
A square has these properties:
- All 4 sides are equal in length
- All 4 angles are right angles (90°)
- The side length is usually called a (or s)
Since a square has all equal sides, its area is simply the side multiplied by itself:
- Area = side × side
- Area = side²
Square units:
- If the side is in cm, area is in cm² (square centimetres)
- If the side is in m, area is in m² (square metres)
- If the side is in km, area is in km² (square kilometres)
Area of Square Formula
Formula:
Area of Square = side × side = a²
Where:
- a = length of one side of the square
- Area is always in square units
Finding the side when area is known:
Side = √Area
For example:
- If Area = 49 cm², then side = √49 = 7 cm
- If Area = 100 m², then side = √100 = 10 m
Relation between area and perimeter of a square:
- Perimeter = 4 × side → side = Perimeter / 4
- Area = side² = (Perimeter / 4)²
Derivation and Proof
The area of a square can be understood by counting unit squares.
Take a square with side 1 cm. Its area is 1 cm × 1 cm = 1 cm². This small square is called a unit square.
Now take a square with side 4 cm. Draw a grid inside it:
- Along the bottom side, you can fit 4 unit squares.
- Along the left side, you can fit 4 unit squares going up.
- Total unit squares = 4 rows × 4 columns = 16 unit squares.
- Area = 16 cm² = 4 × 4 = 4².
This works for any side length:
- Side 5 cm → 5 × 5 = 25 unit squares → Area = 25 cm²
- Side 10 cm → 10 × 10 = 100 unit squares → Area = 100 cm²
The area of a square is a special case of the area of a rectangle. A rectangle's area = length × breadth. In a square, length = breadth = a. So area = a × a = a².
Important: Do not confuse area with perimeter.
- Perimeter of a square = 4a (measured in cm, m)
- Area of a square = a² (measured in cm², m²)
Types and Properties
Type 1: Find the area given the side
- Directly use Area = a².
- Example: Side = 8 cm → Area = 8² = 64 cm²
Type 2: Find the side given the area
- Use side = √Area.
- Example: Area = 81 cm² → side = √81 = 9 cm
Type 3: Find area when perimeter is given
- First find side = Perimeter ÷ 4, then use Area = side².
- Example: Perimeter = 48 cm → side = 12 cm → Area = 144 cm²
Type 4: Word problems (flooring, tiling, painting)
- Find how many tiles fit in a given area, or the cost of painting a square surface.
Type 5: Comparing areas
- Compare areas of two squares with different sides.
- If side is doubled, area becomes 4 times (because (2a)² = 4a²).
Solved Examples
Example 1: Example 1: Finding the Area
Problem: Find the area of a square with side 13 cm.
Solution:
Given:
- Side (a) = 13 cm
Using the formula:
- Area = a²
- Area = 13 × 13
- Area = 169 cm²
Answer: The area is 169 cm².
Example 2: Example 2: Finding the Side from Area
Problem: The area of a square is 225 m². Find the length of its side.
Solution:
Given:
- Area = 225 m²
Using the formula:
- Side = √Area
- Side = √225
- Side = 15 m
Check: 15 × 15 = 225 ✓
Answer: The side is 15 m.
Example 3: Example 3: Finding Area from Perimeter
Problem: The perimeter of a square park is 80 m. Find its area.
Solution:
Step 1: Find the side.
- Perimeter = 4 × side
- 80 = 4 × side
- Side = 80 ÷ 4 = 20 m
Step 2: Find the area.
- Area = side² = 20 × 20 = 400 m²
Answer: The area is 400 m².
Example 4: Example 4: Tiling Problem
Problem: A square room has a side of 6 m. How many square tiles of side 30 cm are needed to cover the floor?
Solution:
Step 1: Convert to the same unit.
- Room side = 6 m = 600 cm
- Tile side = 30 cm
Step 2: Find area of the room.
- Area of room = 600 × 600 = 3,60,000 cm²
Step 3: Find area of one tile.
- Area of tile = 30 × 30 = 900 cm²
Step 4: Number of tiles.
- Number = 3,60,000 ÷ 900 = 400 tiles
Answer: 400 tiles are needed.
Example 5: Example 5: Cost of Painting
Problem: A square wall has a side of 5 m. If painting costs Rs. 12 per square metre, find the total cost.
Solution:
Step 1: Find area.
- Area = 5 × 5 = 25 m²
Step 2: Find cost.
- Cost = Area × Rate
- Cost = 25 × 12 = Rs. 300
Answer: The total cost of painting is Rs. 300.
Example 6: Example 6: Doubling the Side
Problem: A square has side 7 cm. If the side is doubled, how does the area change?
Solution:
Original area:
- Area = 7 × 7 = 49 cm²
New side = 2 × 7 = 14 cm
- New area = 14 × 14 = 196 cm²
Comparison:
- 196 ÷ 49 = 4
Answer: When the side is doubled, the area becomes 4 times the original.
Example 7: Example 7: Finding Perimeter from Area
Problem: The area of a square is 64 cm². Find its perimeter.
Solution:
Step 1: Find the side.
- Side = √64 = 8 cm
Step 2: Find the perimeter.
- Perimeter = 4 × 8 = 32 cm
Answer: The perimeter is 32 cm.
Example 8: Example 8: Word Problem — Garden Bed
Problem: Tina's mother wants to make a square flower bed of side 4 m in their garden. If each square metre needs 6 flower plants, how many plants are needed in total?
Solution:
Step 1: Find area of the flower bed.
- Area = 4 × 4 = 16 m²
Step 2: Find total plants.
- Plants = 16 × 6 = 96 plants
Answer: Tina's mother needs 96 plants.
Example 9: Example 9: Comparing Two Squares
Problem: Square A has side 9 cm. Square B has side 12 cm. How much more area does Square B have?
Solution:
- Area of A = 9 × 9 = 81 cm²
- Area of B = 12 × 12 = 144 cm²
- Difference = 144 − 81 = 63 cm²
Answer: Square B has 63 cm² more area than Square A.
Example 10: Example 10: Land Area in Hectares
Problem: A square piece of land has a side of 100 m. Find its area in square metres and in hectares.
Solution:
Given: 1 hectare = 10,000 m²
- Area = 100 × 100 = 10,000 m²
- Area in hectares = 10,000 ÷ 10,000 = 1 hectare
Answer: The area is 10,000 m² or 1 hectare.
Real-World Applications
The area of a square is used in many real-life situations:
- Flooring and tiling: To find how many tiles are needed to cover a square room, divide the room area by the area of one tile.
- Painting walls: If a wall is square, calculate its area to know how much paint to buy.
- Land measurement: Farmers and builders use area to measure plots of land. A square plot of side 100 m has an area of 1 hectare.
- Making handkerchiefs or napkins: These are usually square. Knowing the area tells you how much cloth is needed.
- Board games: A chessboard is a square made of 64 smaller squares (8 × 8). Each small square's area helps determine the board's total area.
- Solar panels: Square solar panels are measured by area — a larger area captures more sunlight.
Key Points to Remember
- The area of a square = side × side = a².
- Area is measured in square units (cm², m², km²).
- To find the side when area is given: side = √Area.
- If the side of a square is doubled, the area becomes 4 times.
- If the side is tripled, the area becomes 9 times.
- A square is a special rectangle where length = breadth.
- Perimeter (4a) and area (a²) use different units — perimeter is in cm/m, area is in cm²/m².
- From perimeter: side = P/4, then area = (P/4)².
- From area: side = √A, then perimeter = 4√A.
- 1 hectare = 10,000 m² = area of a square with side 100 m.
Practice Problems
- Find the area of a square with side 11 cm.
- The area of a square is 400 m². Find the side.
- The perimeter of a square is 52 cm. Find its area.
- A square playground of side 50 m is to be covered with grass at Rs. 8 per square metre. Find the total cost.
- How many square tiles of side 20 cm are needed to cover a square floor of side 4 m?
- The side of a square is halved. What happens to its area?
- Find the perimeter of a square whose area is 144 cm².
- A square garden has an area of 625 m². Find the cost of fencing it at Rs. 40 per metre.
Frequently Asked Questions
Q1. What is the formula for the area of a square?
Area = side × side = a², where a is the length of one side. For example, if the side is 6 cm, the area = 6 × 6 = 36 cm².
Q2. What is the difference between area and perimeter of a square?
Perimeter is the total length of the boundary (4 × side, measured in cm or m). Area is the space inside the square (side × side, measured in cm² or m²). Perimeter is 1-dimensional, area is 2-dimensional.
Q3. How do you find the side of a square if the area is given?
Take the square root of the area. If Area = 81 cm², then side = √81 = 9 cm.
Q4. What happens to the area if the side of a square is doubled?
The area becomes 4 times the original. If original side = a, new side = 2a. New area = (2a)² = 4a². So doubling the side makes the area 4 times bigger.
Q5. What is a square unit?
A square unit is a square with side 1 unit. For example, 1 cm² is a square with side 1 cm. Area is measured by counting how many such unit squares fit inside the shape.
Q6. Can area and perimeter of a square ever be numerically equal?
Yes, when the side = 4 units. Perimeter = 4 × 4 = 16. Area = 4 × 4 = 16. Both equal 16, but they have different units (cm vs cm²).
Q7. What is a hectare?
A hectare is a unit of area equal to 10,000 square metres. It is the area of a square with side 100 m. Hectares are commonly used for measuring land.
Q8. Is the area of a square the same as the area of a rectangle?
A square is a special rectangle where all sides are equal. The rectangle formula is Area = length × breadth. For a square, length = breadth = side, so Area = side × side = side². The formulas are consistent.
