Relationship Between Area and Perimeter
Many students assume that shapes with the same area must have the same perimeter, or vice versa. This is not true. Area and perimeter are independent measurements — a shape can have a large area with a small perimeter, or a small area with a large perimeter.
Understanding this relationship helps in practical tasks like choosing the best shape for a garden (maximum area with minimum fencing) or designing rooms.
What is Relationship Between Area and Perimeter - Class 4 Maths (Measurement)?
Area = the amount of surface inside a shape (measured in square units).
Perimeter = the distance around the boundary (measured in linear units).
Same perimeter ≠ Same area
Same area ≠ Same perimeter
Key insight: Among all rectangles with the same perimeter, the square has the greatest area. Among all rectangles with the same area, the square has the smallest perimeter.
Solved Examples
Example 1: Example 1: Same Perimeter, Different Area
Problem: Two rectangles each have a perimeter of 20 cm. One is 6 cm x 4 cm and the other is 8 cm x 2 cm. Compare their areas.
Solution:
Rectangle 1: P = 2(6+4) = 20 cm, Area = 6 x 4 = 24 sq cm
Rectangle 2: P = 2(8+2) = 20 cm, Area = 8 x 2 = 16 sq cm
Answer: Same perimeter (20 cm), but areas are different (24 vs 16 sq cm). The shape closer to a square has more area.
Example 2: Example 2: Same Area, Different Perimeter
Problem: Two rectangles each have an area of 24 sq cm. One is 6 cm x 4 cm and the other is 12 cm x 2 cm. Compare their perimeters.
Solution:
Rectangle 1: Area = 24, P = 2(6+4) = 20 cm
Rectangle 2: Area = 24, P = 2(12+2) = 28 cm
Answer: Same area (24 sq cm), but perimeters are different (20 vs 28 cm). The more elongated shape has a larger perimeter.
Example 3: Example 3: Square is Best for Fencing
Problem: Aman has 24 m of fencing. What rectangle gives the maximum area?
Solution:
Perimeter = 24, so Length + Breadth = 12.
Try: 11 x 1 = 11, 10 x 2 = 20, 9 x 3 = 27, 8 x 4 = 32, 7 x 5 = 35, 6 x 6 = 36
Answer: A square of 6 m x 6 m gives the maximum area of 36 sq m.
Example 4: Example 4: Minimum Perimeter for Given Area
Problem: Priya wants to tile 36 sq cm with a rectangular shape. Which rectangle uses the least border tape?
Solution:
Rectangles with area 36: 36x1 (P=74), 18x2 (P=40), 12x3 (P=30), 9x4 (P=26), 6x6 (P=24)
Answer: A 6 x 6 square with perimeter 24 cm uses the least tape.
Example 5: Example 5: Grid Exploration
Problem: On a grid, draw two shapes each with area 8 square units. Compare their perimeters.
Solution:
Shape 1: A 4 x 2 rectangle → Perimeter = 2(4+2) = 12 units
Shape 2: An L-shape of 8 squares → Count outer edges: might be 14 or more units
Answer: Both have area 8, but the L-shape has a larger perimeter. More "bumpy" shapes have larger perimeters.
Example 6: Example 6: Word Problem — Garden
Problem: Meera has a 10 m x 5 m garden and Dev has a 25 m x 2 m garden. Who has more area? Who needs more fencing?
Solution:
Meera: Area = 50 sq m, Perimeter = 30 m
Dev: Area = 50 sq m, Perimeter = 54 m
Answer: Both have the same area (50 sq m), but Dev needs more fencing (54 m vs 30 m).
Example 7: Example 7: Increasing One, Decreasing Other
Problem: A 5 x 5 square has area 25 sq cm and perimeter 20 cm. A 10 x 1 rectangle has area 10 sq cm and perimeter 22 cm. What do you notice?
Solution:
The rectangle has less area (10 < 25) but more perimeter (22 > 20).
Answer: A larger perimeter does not mean a larger area.
Example 8: Example 8: Real-Life Application
Problem: Rahul can buy 20 m of ribbon. He wants to make a rectangular frame with the most space inside. What dimensions should he choose?
Solution:
Perimeter = 20 m, so L + B = 10.
Best choice: 5 m x 5 m (square) → Area = 25 sq m.
Answer: A 5 m x 5 m square frame has the most space.
Example 9: Example 9: Same Perimeter Table
Problem: List all rectangles (whole number sides) with perimeter 16 cm and their areas.
Solution:
| L | B | Perimeter | Area |
|---|---|---|---|
| 7 | 1 | 16 | 7 |
| 6 | 2 | 16 | 12 |
| 5 | 3 | 16 | 15 |
| 4 | 4 | 16 | 16 |
Answer: The square (4 x 4) has the largest area (16 sq cm).
Key Points to Remember
- Area and perimeter are different measures — they do not depend on each other.
- Shapes with the same perimeter can have different areas.
- Shapes with the same area can have different perimeters.
- Among rectangles with the same perimeter, the square has the maximum area.
- Among rectangles with the same area, the square has the minimum perimeter.
- More "stretched out" or "bumpy" shapes tend to have larger perimeters for the same area.
Practice Problems
- Draw two rectangles with perimeter 24 cm. Calculate the area of each.
- Draw two rectangles with area 20 sq cm. Calculate the perimeter of each.
- Aditi has 28 m of fencing. What square dimensions give maximum area?
- A 12 x 3 rectangle and a 6 x 6 square — which has more area? Which has more perimeter?
- List all rectangles (whole number sides) with area 18 sq cm. Which has the smallest perimeter?
- True or False: If you double the perimeter of a rectangle, the area also doubles.
Frequently Asked Questions
Q1. Can two shapes have the same perimeter but different areas?
Yes. For example, a 6x4 rectangle (perimeter 20, area 24) and a 9x1 rectangle (perimeter 20, area 9) have the same perimeter but very different areas.
Q2. Can two shapes have the same area but different perimeters?
Yes. A 6x6 square (area 36, perimeter 24) and a 18x2 rectangle (area 36, perimeter 40) have the same area but different perimeters.
Q3. Which shape gives maximum area for a given perimeter?
Among rectangles, a square gives the maximum area. For example, with perimeter 20 cm, a 5x5 square gives area 25 sq cm — more than any other rectangle.
Q4. Does a bigger perimeter always mean bigger area?
No. A long thin shape can have a large perimeter but a small area. Perimeter and area are independent measurements.
Q5. How are area and perimeter different?
Area measures the surface inside a shape (in square units). Perimeter measures the distance around the boundary (in linear units like cm or m).
Q6. If you double the sides of a square, what happens to area and perimeter?
The perimeter doubles (from 4s to 8s). The area quadruples (from s² to 4s²). They do not change proportionally.
Q7. Why is this relationship useful?
In real life, you may want to maximize the garden area with limited fencing, or minimize material for a given floor space. Understanding this relationship helps make efficient choices.
Q8. Is this topic in NCERT Class 4?
Yes, exploring the relationship between area and perimeter is part of the CBSE/NCERT Class 4 Maths curriculum under the Measurement chapter.
