Chapter 11: Area and its Boundary
5th StandardMathematics
Chapter Summary
Area and its Boundary - Chapter Summary
# Area and Its Boundary
## Overview
This chapter introduces the concept of area and perimeter (boundary) through practical activities and relatable stories. Students learn to compare areas of different shapes, measure perimeter using simple tools, estimate space using square units, and understand how different shapes with the same area can have different boundaries.
## Key Topics Covered
### 1. Comparing Areas Using Squares
* **Scenario**: Parth and Gini compare two pieces of aam paapad using square units.
* **Activity**: Use 1 cm × 1 cm squares to cover the shapes.
* **Learning**:
* Count the number of squares to find the area.
* Area = length × width (for rectangles).
* Compare areas to decide which is larger.
### 2. Estimating Area Using Objects
* **Stamps and Shapes**: Use stamps of known area (e.g., 4 cm²) to cover larger rectangles.
* **Process**:
* Estimate how many stamps cover a surface.
* Confirm by measuring length and width.
* Multiply to find area.
* **Extension**: Also compute perimeter (sum of all side lengths).
### 3. Real-life Applications
* **Tiling the Floor**:
* Find how many 10 cm tiles fit in a kitchen of size 220 cm × 180 cm.
* **Fencing a Garden**:
* Given perimeter of a square garden, find side length.
* **Wire for a Rectangle**:
* Use given perimeter and one dimension to find the other.
### 4. Understanding Area and Perimeter Together
* **Greeting Cards Table**: Fill in missing area/perimeter using known dimensions.
* **Make Belts**:
* Cut strips from a postcard to make belts of equal area but different perimeters.
* Learn: Different shapes with same area can have different boundaries.
### 5. One Square Metre Activity
* **Team Game**:
* Make 1 m² squares and see how many people can sit/stand in them.
* Understand how square metre measures space.
* **Classroom Floor**:
* Measure length and width of classroom.
* Multiply to find area in m².
* Estimate how many students can fit comfortably.
### 6. Land Sharing and Population Density
* **Sharing Land**:
* Divide a plot among three children, each getting a tree.
* Estimate area using square metres.
* Calculate fencing required for each plot.
* **Population in a Square Kilometre**:
* Compare densely (West Bengal) vs. sparsely (Arunachal Pradesh) populated areas.
* Sense of size of 1 km² introduced through maps.
### 7. Drawing and Understanding Squares
* **Square Comparison**:
* Draw squares A and B with side lengths doubled.
* Discover that area increases 4 times, but perimeter only 2 times.
### 8. Thread Play
* **Activity**:
* Make different shapes with the same length of thread (15 cm).
* Compare which has the largest or smallest area.
* **Shapes**: Triangle, square, rectangle, circle.
### 9. Save the Birds (Perimeter in Maps)
* **Map Activity**:
* Measure lake boundaries using thread and convert to km.
* Decide which lake to keep quiet for birds (based on larger boundary).
* Measure area of lake from drawing to estimate space for birds.
### 10. King’s Story (Maximising Area with Fixed Perimeter)
* **Story**:
* Carpenters Cheggu and Anar rewarded with land/gold wire.
* **Exploration**:
* Cheggu makes different rectangles with same perimeter (100 m) to get larger area.
* Anar makes rectangles of area 800 m² with different perimeters, sees how boundary length changes.
* Learns that longer, thinner shapes need more boundary for same area.
---
## New Terms and Simple Definitions
| Term | Simple Definition |
| ---------------- | ---------------------------------------------------------------------------- |
| Area | The amount of space inside a shape, measured in square units (like cm², m²). |
| Perimeter | The total length around a shape – add all the sides. |
| Square Unit | A square of 1 unit length and 1 unit width (like 1 cm × 1 cm = 1 cm²). |
| Boundary | The edge or outline of a shape. |
| Estimate | A close guess or idea of a number or amount. |
| Rectangle | A shape with four sides and four right angles; opposite sides are equal. |
| Postcard Loop | A loop made by cutting and folding a postcard in a special way. |
| Square Metre | A square that is 1 metre on each side (1 m × 1 m = 1 m²). |
| Square Kilometre | A big square that is 1 kilometre on each side (1 km × 1 km = 1 km²). |
| Wire (in Math) | A string or rope used to measure the boundary of shapes. |
---
## Practice Time
Here are 10 practice sums of varying difficulty levels:
### Easy (3)
1. **Find the area of a rectangle with length 7 cm and width 4 cm.**
* Area = 7 × 4 = **28 cm²**
2. **What is the perimeter of a square with each side 5 cm?**
* Perimeter = 4 × 5 = **20 cm**
3. **A tile is 10 cm long and 10 cm wide. What is its area?**
* Area = 10 × 10 = **100 cm²**
### Medium (2)
4. **A rectangular garden is 12 m long and 8 m wide. What is its perimeter?**
* Perimeter = 2 × (12 + 8) = 2 × 20 = **40 m**
5. **How many square tiles of side 20 cm are needed to cover a floor of size 200 cm × 180 cm?**
* Area of floor = 200 × 180 = 36,000 cm²
* Area of 1 tile = 20 × 20 = 400 cm²
* Number of tiles = 36,000 ÷ 400 = **90 tiles**
### Difficult (3)
6. **A wire is 24 metres long. It is bent into a rectangle where the width is 5 m. What is the length and area?**
* Perimeter = 2 × (L + 5) = 24 → L + 5 = 12 → L = 7
* Area = 7 × 5 = **35 m²**
7. **If the area of a square is 144 m², what is the length of one side and its perimeter?**
* Side = √144 = 12 m
* Perimeter = 4 × 12 = **48 m**
8. **You draw a square of side 3 cm and another of side 6 cm. How many times larger is the area of the second square?**
* Area1 = 3 × 3 = 9 cm²
* Area2 = 6 × 6 = 36 cm²
* Ratio = 36 ÷ 9 = **4 times larger**
### Very Difficult (2)
9. **A rectangle has area 800 m². What are possible pairs of (length, width) with integer values that give this area? Which gives the smallest perimeter?**
* (1, 800), (2, 400), (4, 200), (10, 80), (20, 40)
* Smallest perimeter = 2 × (20 + 40) = **120 m**
10. **A thread of 15 cm is used to form different shapes. Which shape will have the maximum area – square, rectangle, or circle?**
* For fixed perimeter, **circle** gives maximum area.
---
## Overview
This chapter introduces the concept of area and perimeter (boundary) through practical activities and relatable stories. Students learn to compare areas of different shapes, measure perimeter using simple tools, estimate space using square units, and understand how different shapes with the same area can have different boundaries.
## Key Topics Covered
### 1. Comparing Areas Using Squares
* **Scenario**: Parth and Gini compare two pieces of aam paapad using square units.
* **Activity**: Use 1 cm × 1 cm squares to cover the shapes.
* **Learning**:
* Count the number of squares to find the area.
* Area = length × width (for rectangles).
* Compare areas to decide which is larger.
### 2. Estimating Area Using Objects
* **Stamps and Shapes**: Use stamps of known area (e.g., 4 cm²) to cover larger rectangles.
* **Process**:
* Estimate how many stamps cover a surface.
* Confirm by measuring length and width.
* Multiply to find area.
* **Extension**: Also compute perimeter (sum of all side lengths).
### 3. Real-life Applications
* **Tiling the Floor**:
* Find how many 10 cm tiles fit in a kitchen of size 220 cm × 180 cm.
* **Fencing a Garden**:
* Given perimeter of a square garden, find side length.
* **Wire for a Rectangle**:
* Use given perimeter and one dimension to find the other.
### 4. Understanding Area and Perimeter Together
* **Greeting Cards Table**: Fill in missing area/perimeter using known dimensions.
* **Make Belts**:
* Cut strips from a postcard to make belts of equal area but different perimeters.
* Learn: Different shapes with same area can have different boundaries.
### 5. One Square Metre Activity
* **Team Game**:
* Make 1 m² squares and see how many people can sit/stand in them.
* Understand how square metre measures space.
* **Classroom Floor**:
* Measure length and width of classroom.
* Multiply to find area in m².
* Estimate how many students can fit comfortably.
### 6. Land Sharing and Population Density
* **Sharing Land**:
* Divide a plot among three children, each getting a tree.
* Estimate area using square metres.
* Calculate fencing required for each plot.
* **Population in a Square Kilometre**:
* Compare densely (West Bengal) vs. sparsely (Arunachal Pradesh) populated areas.
* Sense of size of 1 km² introduced through maps.
### 7. Drawing and Understanding Squares
* **Square Comparison**:
* Draw squares A and B with side lengths doubled.
* Discover that area increases 4 times, but perimeter only 2 times.
### 8. Thread Play
* **Activity**:
* Make different shapes with the same length of thread (15 cm).
* Compare which has the largest or smallest area.
* **Shapes**: Triangle, square, rectangle, circle.
### 9. Save the Birds (Perimeter in Maps)
* **Map Activity**:
* Measure lake boundaries using thread and convert to km.
* Decide which lake to keep quiet for birds (based on larger boundary).
* Measure area of lake from drawing to estimate space for birds.
### 10. King’s Story (Maximising Area with Fixed Perimeter)
* **Story**:
* Carpenters Cheggu and Anar rewarded with land/gold wire.
* **Exploration**:
* Cheggu makes different rectangles with same perimeter (100 m) to get larger area.
* Anar makes rectangles of area 800 m² with different perimeters, sees how boundary length changes.
* Learns that longer, thinner shapes need more boundary for same area.
---
## New Terms and Simple Definitions
| Term | Simple Definition |
| ---------------- | ---------------------------------------------------------------------------- |
| Area | The amount of space inside a shape, measured in square units (like cm², m²). |
| Perimeter | The total length around a shape – add all the sides. |
| Square Unit | A square of 1 unit length and 1 unit width (like 1 cm × 1 cm = 1 cm²). |
| Boundary | The edge or outline of a shape. |
| Estimate | A close guess or idea of a number or amount. |
| Rectangle | A shape with four sides and four right angles; opposite sides are equal. |
| Postcard Loop | A loop made by cutting and folding a postcard in a special way. |
| Square Metre | A square that is 1 metre on each side (1 m × 1 m = 1 m²). |
| Square Kilometre | A big square that is 1 kilometre on each side (1 km × 1 km = 1 km²). |
| Wire (in Math) | A string or rope used to measure the boundary of shapes. |
---
## Practice Time
Here are 10 practice sums of varying difficulty levels:
### Easy (3)
1. **Find the area of a rectangle with length 7 cm and width 4 cm.**
* Area = 7 × 4 = **28 cm²**
2. **What is the perimeter of a square with each side 5 cm?**
* Perimeter = 4 × 5 = **20 cm**
3. **A tile is 10 cm long and 10 cm wide. What is its area?**
* Area = 10 × 10 = **100 cm²**
### Medium (2)
4. **A rectangular garden is 12 m long and 8 m wide. What is its perimeter?**
* Perimeter = 2 × (12 + 8) = 2 × 20 = **40 m**
5. **How many square tiles of side 20 cm are needed to cover a floor of size 200 cm × 180 cm?**
* Area of floor = 200 × 180 = 36,000 cm²
* Area of 1 tile = 20 × 20 = 400 cm²
* Number of tiles = 36,000 ÷ 400 = **90 tiles**
### Difficult (3)
6. **A wire is 24 metres long. It is bent into a rectangle where the width is 5 m. What is the length and area?**
* Perimeter = 2 × (L + 5) = 24 → L + 5 = 12 → L = 7
* Area = 7 × 5 = **35 m²**
7. **If the area of a square is 144 m², what is the length of one side and its perimeter?**
* Side = √144 = 12 m
* Perimeter = 4 × 12 = **48 m**
8. **You draw a square of side 3 cm and another of side 6 cm. How many times larger is the area of the second square?**
* Area1 = 3 × 3 = 9 cm²
* Area2 = 6 × 6 = 36 cm²
* Ratio = 36 ÷ 9 = **4 times larger**
### Very Difficult (2)
9. **A rectangle has area 800 m². What are possible pairs of (length, width) with integer values that give this area? Which gives the smallest perimeter?**
* (1, 800), (2, 400), (4, 200), (10, 80), (20, 40)
* Smallest perimeter = 2 × (20 + 40) = **120 m**
10. **A thread of 15 cm is used to form different shapes. Which shape will have the maximum area – square, rectangle, or circle?**
* For fixed perimeter, **circle** gives maximum area.
---
Area and Its Boundary
Overview
This chapter introduces the concept of area and perimeter (boundary) through practical activities and relatable stories. Students learn to compare areas of different shapes, measure perimeter using simple tools, estimate space using square units, and understand how different shapes with the same area can have different boundaries.
Key Topics Covered
1. Comparing Areas Using Squares
-
Scenario: Parth and Gini compare two pieces of aam paapad using square units.
-
Activity: Use 1 cm × 1 cm squares to cover the shapes.
-
Learning:
- Count the number of squares to find the area.
- Area = length × width (for rectangles).
- Compare areas to decide which is larger.
2. Estimating Area Using Objects
-
Stamps and Shapes: Use stamps of known area (e.g., 4 cm²) to cover larger rectangles.
-
Process:
- Estimate how many stamps cover a surface.
- Confirm by measuring length and width.
- Multiply to find area.
-
Extension: Also compute perimeter (sum of all side lengths).
3. Real-life Applications
-
Tiling the Floor:
- Find how many 10 cm tiles fit in a kitchen of size 220 cm × 180 cm.
-
Fencing a Garden:
- Given perimeter of a square garden, find side length.
-
Wire for a Rectangle:
- Use given perimeter and one dimension to find the other.
4. Understanding Area and Perimeter Together
-
Greeting Cards Table: Fill in missing area/perimeter using known dimensions.
-
Make Belts:
- Cut strips from a postcard to make belts of equal area but different perimeters.
- Learn: Different shapes with same area can have different boundaries.
5. One Square Metre Activity
-
Team Game:
- Make 1 m² squares and see how many people can sit/stand in them.
- Understand how square metre measures space.
-
Classroom Floor:
- Measure length and width of classroom.
- Multiply to find area in m².
- Estimate how many students can fit comfortably.
6. Land Sharing and Population Density
-
Sharing Land:
- Divide a plot among three children, each getting a tree.
- Estimate area using square metres.
- Calculate fencing required for each plot.
-
Population in a Square Kilometre:
- Compare densely (West Bengal) vs. sparsely (Arunachal Pradesh) populated areas.
- Sense of size of 1 km² introduced through maps.
7. Drawing and Understanding Squares
-
Square Comparison:
- Draw squares A and B with side lengths doubled.
- Discover that area increases 4 times, but perimeter only 2 times.
8. Thread Play
-
Activity:
- Make different shapes with the same length of thread (15 cm).
- Compare which has the largest or smallest area.
-
Shapes: Triangle, square, rectangle, circle.
9. Save the Birds (Perimeter in Maps)
-
Map Activity:
- Measure lake boundaries using thread and convert to km.
- Decide which lake to keep quiet for birds (based on larger boundary).
- Measure area of lake from drawing to estimate space for birds.
10. King’s Story (Maximising Area with Fixed Perimeter)
-
Story:
- Carpenters Cheggu and Anar rewarded with land/gold wire.
-
Exploration:
- Cheggu makes different rectangles with same perimeter (100 m) to get larger area.
- Anar makes rectangles of area 800 m² with different perimeters, sees how boundary length changes.
- Learns that longer, thinner shapes need more boundary for same area.
New Terms and Simple Definitions
| Term | Simple Definition |
|---|---|
| Area | The amount of space inside a shape, measured in square units (like cm², m²). |
| Perimeter | The total length around a shape – add all the sides. |
| Square Unit | A square of 1 unit length and 1 unit width (like 1 cm × 1 cm = 1 cm²). |
| Boundary | The edge or outline of a shape. |
| Estimate | A close guess or idea of a number or amount. |
| Rectangle | A shape with four sides and four right angles; opposite sides are equal. |
| Postcard Loop | A loop made by cutting and folding a postcard in a special way. |
| Square Metre | A square that is 1 metre on each side (1 m × 1 m = 1 m²). |
| Square Kilometre | A big square that is 1 kilometre on each side (1 km × 1 km = 1 km²). |
| Wire (in Math) | A string or rope used to measure the boundary of shapes. |
Practice Time
Here are 10 practice sums of varying difficulty levels:
Easy (3)
-
Find the area of a rectangle with length 7 cm and width 4 cm.
- Area = 7 × 4 = 28 cm²
-
What is the perimeter of a square with each side 5 cm?
- Perimeter = 4 × 5 = 20 cm
-
A tile is 10 cm long and 10 cm wide. What is its area?
- Area = 10 × 10 = 100 cm²
Medium (2)
-
A rectangular garden is 12 m long and 8 m wide. What is its perimeter?
- Perimeter = 2 × (12 + 8) = 2 × 20 = 40 m
-
How many square tiles of side 20 cm are needed to cover a floor of size 200 cm × 180 cm?
- Area of floor = 200 × 180 = 36,000 cm²
- Area of 1 tile = 20 × 20 = 400 cm²
- Number of tiles = 36,000 ÷ 400 = 90 tiles
Difficult (3)
-
A wire is 24 metres long. It is bent into a rectangle where the width is 5 m. What is the length and area?
- Perimeter = 2 × (L + 5) = 24 → L + 5 = 12 → L = 7
- Area = 7 × 5 = 35 m²
-
If the area of a square is 144 m², what is the length of one side and its perimeter?
- Side = √144 = 12 m
- Perimeter = 4 × 12 = 48 m
-
You draw a square of side 3 cm and another of side 6 cm. How many times larger is the area of the second square?
- Area1 = 3 × 3 = 9 cm²
- Area2 = 6 × 6 = 36 cm²
- Ratio = 36 ÷ 9 = 4 times larger
Very Difficult (2)
-
A rectangle has area 800 m². What are possible pairs of (length, width) with integer values that give this area? Which gives the smallest perimeter?
- (1, 800), (2, 400), (4, 200), (10, 80), (20, 40)
- Smallest perimeter = 2 × (20 + 40) = 120 m
-
A thread of 15 cm is used to form different shapes. Which shape will have the maximum area – square, rectangle, or circle?
- For fixed perimeter, circle gives maximum area.