Area of Rectangle
Imagine you want to cover the floor of your room with tiles. You need to know how much space the floor takes up. The amount of flat space inside a shape is called its area.
A rectangle is a shape with 4 sides and 4 right angles (90 degree corners). The opposite sides are equal. The longer side is usually called the length and the shorter side is called the breadth (or width).
The area of a rectangle tells us how many square units fit inside it. If you draw a rectangle on graph paper, you can count the squares inside — that count is the area.
In this topic, you will learn the formula for the area of a rectangle, how to use it, and how to solve word problems involving area.
What is Area of Rectangle - Grade 6 Maths (Mensuration)?
Definition: The area of a rectangle is the amount of surface enclosed within its four sides. It is measured in square units (like cm², m², km²).
Key terms:
- Length (l): The longer side of the rectangle.
- Breadth (b): The shorter side of the rectangle (also called width).
- Square unit: A unit of area. One square centimetre (1 cm²) is the area of a square with side 1 cm.
Important:
- Area is always in square units (cm², m², km²), never in plain cm or m.
- Both length and breadth must be in the same unit before calculating area.
- Area tells us "how much space" — it is different from perimeter, which tells us "how much boundary."
Area of Rectangle Formula
Formula:
Area of Rectangle = length × breadth = l × b
Where:
- l = length of the rectangle
- b = breadth (width) of the rectangle
- Area is in square units
Related formulas:
Length = Area ÷ Breadth
Breadth = Area ÷ Length
These are used when one dimension is unknown.
Derivation and Proof
Understanding the formula through counting squares:
Draw a rectangle with length 5 cm and breadth 3 cm on graph paper (each small square = 1 cm²).
Step 1: Count the squares along the length. There are 5 squares in each row.
Step 2: Count the number of rows (along the breadth). There are 3 rows.
Step 3: Total squares = 5 × 3 = 15 squares.
Step 4: Each square = 1 cm². So the area = 15 cm².
This is exactly length × breadth = 5 × 3 = 15 cm².
The formula works because we are counting how many unit squares fit inside the rectangle. The number of squares in each row equals the length, and the number of rows equals the breadth. So the total is always length × breadth.
Unit conversion reminder:
- 1 m = 100 cm → 1 m² = 10,000 cm²
- 1 km = 1000 m → 1 km² = 1,000,000 m²
- Always convert to the same unit before multiplying.
Types and Properties
Types of Area Problems:
Type 1: Finding Area (both sides given)
- Given length and breadth, find area = l × b
- Example: l = 12 cm, b = 7 cm → Area = 84 cm²
Type 2: Finding a Missing Side (area and one side given)
- Given area and one side, find the other
- Example: Area = 60 cm², l = 10 cm → b = 60 ÷ 10 = 6 cm
Type 3: Unit Conversion Problems
- Sides given in different units — convert first, then calculate
- Example: l = 2 m, b = 50 cm → convert to same unit → l = 200 cm, b = 50 cm → Area = 10,000 cm²
Type 4: Word Problems
- Real-life problems involving floors, walls, gardens, fields, etc.
- Read carefully to identify length and breadth from the problem.
Type 5: Comparing Areas
- Compare the areas of two rectangles to find which is larger.
Solved Examples
Example 1: Example 1: Basic Area Calculation
Problem: Find the area of a rectangle with length 14 cm and breadth 9 cm.
Solution:
Given:
- Length (l) = 14 cm
- Breadth (b) = 9 cm
Using the formula:
- Area = l × b
- Area = 14 × 9
- Area = 126 cm²
Answer: The area is 126 cm².
Example 2: Example 2: Finding Breadth from Area
Problem: The area of a rectangle is 72 m² and its length is 12 m. Find the breadth.
Solution:
Given:
- Area = 72 m²
- Length (l) = 12 m
Using the formula:
- Breadth = Area ÷ Length
- Breadth = 72 ÷ 12
- Breadth = 6 m
Answer: The breadth is 6 m.
Example 3: Example 3: Area with Unit Conversion
Problem: A table top is 1.5 m long and 80 cm wide. Find its area in cm².
Solution:
Given:
- Length = 1.5 m = 150 cm
- Breadth = 80 cm
Using the formula:
- Area = l × b
- Area = 150 × 80
- Area = 12,000 cm²
Answer: The area is 12,000 cm².
Example 4: Example 4: Area of a Room Floor
Problem: A room is 8 m long and 6 m wide. Find the cost of tiling the floor at Rs 45 per m².
Solution:
Given:
- Length = 8 m, Breadth = 6 m
- Cost per m² = Rs 45
Step 1: Find the area.
- Area = 8 × 6 = 48 m²
Step 2: Find the total cost.
- Cost = Area × Rate = 48 × 45 = Rs 2,160
Answer: The cost of tiling is Rs 2,160.
Example 5: Example 5: Comparing Two Rectangles
Problem: Rectangle A has length 15 cm and breadth 8 cm. Rectangle B has length 12 cm and breadth 11 cm. Which has a greater area?
Solution:
- Area of A = 15 × 8 = 120 cm²
- Area of B = 12 × 11 = 132 cm²
- 132 > 120
Answer: Rectangle B has a greater area (132 cm² > 120 cm²).
Example 6: Example 6: Finding Length from Area
Problem: A rectangular garden has area 180 m² and breadth 9 m. Find the length.
Solution:
- Length = Area ÷ Breadth
- Length = 180 ÷ 9
- Length = 20 m
Answer: The length is 20 m.
Example 7: Example 7: Number of Tiles
Problem: A floor is 6 m long and 4 m wide. How many tiles of size 50 cm × 50 cm are needed to cover it?
Solution:
Step 1: Find the area of the floor.
- Floor area = 6 × 4 = 24 m² = 240,000 cm²
Step 2: Find the area of one tile.
- Tile area = 50 × 50 = 2,500 cm²
Step 3: Number of tiles = Floor area ÷ Tile area
- = 240,000 ÷ 2,500 = 96 tiles
Answer: 96 tiles are needed.
Example 8: Example 8: Area of a Path
Problem: A garden is 20 m long and 15 m wide. A path 2 m wide runs inside along all four sides. Find the area of the path.
Solution:
Step 1: Area of outer rectangle (garden) = 20 × 15 = 300 m²
Step 2: Inner rectangle dimensions:
- Inner length = 20 − 2 − 2 = 16 m
- Inner breadth = 15 − 2 − 2 = 11 m
Step 3: Area of inner rectangle = 16 × 11 = 176 m²
Step 4: Area of path = Outer area − Inner area = 300 − 176 = 124 m²
Answer: The area of the path is 124 m².
Example 9: Example 9: Area in Hectares
Problem: A field is 500 m long and 200 m wide. Find its area in hectares. (1 hectare = 10,000 m²)
Solution:
- Area = 500 × 200 = 100,000 m²
- Area in hectares = 100,000 ÷ 10,000 = 10 hectares
Answer: The area of the field is 10 hectares.
Example 10: Example 10: Doubling the Sides
Problem: A rectangle has length 6 cm and breadth 4 cm. If both sides are doubled, how does the area change?
Solution:
- Original area = 6 × 4 = 24 cm²
- New length = 12 cm, New breadth = 8 cm
- New area = 12 × 8 = 96 cm²
- Change: 96 ÷ 24 = 4
Answer: The area becomes 4 times the original. When both sides are doubled, the area becomes 4 times.
Real-World Applications
Area of rectangle in real life:
- Flooring: To find how many tiles are needed for a room, calculate the floor area and divide by the tile area.
- Painting walls: The area of a wall tells you how much paint to buy.
- Farming: Farmers calculate the area of their fields to know how many seeds to plant or how much fertiliser to use.
- Buying fabric: When buying cloth for curtains or a tablecloth, you need to know the area to cover.
- Sports grounds: The area of a basketball court, football field, or cricket pitch is calculated using the rectangle formula.
- Construction: Builders use area to calculate the amount of material needed for floors, roofs, and walls.
Key Points to Remember
- Area = length × breadth (l × b).
- Area is measured in square units (cm², m², km²).
- Both length and breadth must be in the same unit before multiplying.
- Length = Area ÷ Breadth. Breadth = Area ÷ Length.
- When both sides are doubled, the area becomes 4 times.
- When both sides are tripled, the area becomes 9 times.
- A square is a special rectangle where l = b, so Area of square = side × side = side².
- Do not confuse area (space inside) with perimeter (boundary length).
- 1 m² = 10,000 cm². 1 km² = 1,000,000 m². 1 hectare = 10,000 m².
- Always write the unit as squared (cm², not cm).
Practice Problems
- Find the area of a rectangle with length 25 cm and breadth 16 cm.
- A rectangular plot is 30 m long and 18 m wide. Find its area.
- The area of a rectangle is 96 cm² and its breadth is 8 cm. Find the length.
- How many tiles of size 20 cm × 20 cm are needed to cover a floor of 4 m × 3 m?
- A rectangular park is 120 m by 80 m. Find the cost of planting grass at Rs 12 per m².
- The length of a rectangle is three times its breadth. If the breadth is 7 cm, find the area.
- A room is 5 m long and 4 m wide. A carpet covers the entire floor. Find the area of the carpet in cm².
- Two rectangles have the same area of 48 cm². One has length 12 cm. The other has length 8 cm. Find the breadth of each.
Frequently Asked Questions
Q1. What is the area of a rectangle?
The area of a rectangle is the space enclosed within its four sides. It is calculated by multiplying the length by the breadth: Area = l × b. The answer is always in square units.
Q2. What is the difference between area and perimeter?
Area is the space inside a shape, measured in square units (like cm²). Perimeter is the total length of the boundary, measured in normal units (like cm). For a rectangle: Area = l × b, Perimeter = 2(l + b).
Q3. Why is area measured in square units?
Because area measures how many unit squares (like 1 cm × 1 cm squares) fit inside a shape. When you multiply length (in cm) by breadth (in cm), you get cm × cm = cm² (square centimetres).
Q4. Can two rectangles have the same area but different perimeters?
Yes. For example, a 12 × 3 rectangle and a 6 × 6 square both have area 36 cm². But their perimeters are different: 2(12 + 3) = 30 cm and 2(6 + 6) = 24 cm.
Q5. What happens to the area if only the length is doubled?
If only the length is doubled (and breadth stays the same), the area also doubles. For example, a 5 × 4 rectangle has area 20. If length becomes 10 × 4, the area becomes 40 (doubled).
Q6. How do I convert m² to cm²?
Multiply by 10,000. Since 1 m = 100 cm, we get 1 m² = 100 × 100 = 10,000 cm². So 3 m² = 30,000 cm².
Q7. Is the area formula the same for a square?
A square is a special rectangle where length = breadth = side. So Area = side × side = side². The rectangle formula still works: l × b = s × s = s².
Q8. How do I find the area if sides are in different units?
Convert both sides to the same unit first, then multiply. For example, if length = 2 m and breadth = 50 cm, convert length to 200 cm. Then area = 200 × 50 = 10,000 cm².
Q9. What is a hectare?
A hectare is a unit of area used for measuring large fields and plots. 1 hectare = 10,000 m². A square field of 100 m × 100 m has an area of 1 hectare.
Q10. Can the area of a rectangle be less than its perimeter?
Yes, but they measure different things (area in square units, perimeter in linear units), so comparing them directly does not have mathematical meaning. However, numerically, a 1 × 2 rectangle has area 2 and perimeter 6.
