Perimeter and Area
1. What Is Perimeter?
PERIMETER is the TOTAL DISTANCE around the OUTSIDE of a shape.
'Think of walking AROUND a playground. The distance you walk is the PERIMETER.'
Key Fact:
'Perimeter is measured in units of LENGTH — cm, m, km, etc.'
How to Find Perimeter:
Add the lengths of ALL sides of the shape.
2. Perimeter of Common Shapes
Square:
All FOUR sides are EQUAL.
Formula: Perimeter = 4 × Side
Example: Find perimeter of a square with side = 5 cm.
Perimeter = 4 × 5 = 20 cm
| Side | Perimeter |
|---|---|
| 3 cm | 12 cm |
| 7 cm | 28 cm |
| 12 m | 48 m |
| 25 mm | 100 mm |
Rectangle:
OPPOSITE sides are EQUAL.
Formula: Perimeter = 2 × (Length + Breadth) Alternatively: Perimeter = 2 × L + 2 × B
Example: Find perimeter of a rectangle with L = 8 cm, B = 3 cm.
Perimeter = 2 × (8 + 3) = 2 × 11 = 22 cm
| Length | Breadth | Perimeter |
|---|---|---|
| 6 cm | 4 cm | 20 cm |
| 10 m | 7 m | 34 m |
| 15 cm | 12 cm | 54 cm |
Triangle:
A triangle has THREE sides.
Formula: Perimeter = Side₁ + Side₂ + Side₃
Example: Find perimeter of a triangle with sides 5 cm, 7 cm, 9 cm.
Perimeter = 5 + 7 + 9 = 21 cm
| Side 1 | Side 2 | Side 3 | Perimeter |
|---|---|---|---|
| 3 cm | 4 cm | 5 cm | 12 cm |
| 6 m | 8 m | 10 m | 24 m |
| 12 cm | 12 cm | 12 cm | 36 cm |
'For an EQUILATERAL triangle (all sides equal), Perimeter = 3 × Side.'
3. Real-Life Perimeter Problems
Problem 1 — Fencing:
A square garden has a side of 12 m. How much fencing wire is needed to surround it?
Solution: Perimeter = 4 × 12 = 48 m of wire.
Problem 2 — Border:
A rectangular photo frame is 25 cm long and 18 cm wide. What is its perimeter?
Solution: Perimeter = 2 × (25 + 18) = 2 × 43 = 86 cm.
Problem 3 — Track:
A triangular park has sides of 50 m, 65 m, and 70 m. What distance is covered in one round?
Solution: Perimeter = 50 + 65 + 70 = 185 m per round.
4. What Is Area?
AREA is the amount of SPACE INSIDE a shape.
'Think of the floor inside a room. The number of tiles needed to cover it is the AREA.'
Key Fact:
'Area is measured in SQUARE units — sq cm (cm²), sq m (m²), etc.'
5. Finding Area Using a Grid
Count the number of SQUARES inside the shape.
Example: A rectangle on a grid that is 6 squares long and 4 squares wide.
Area = Number of squares = 6 × 4 = 24 square units.
| Shape | Grid Size | Area (in squares) |
|---|---|---|
| Square (3 × 3) | 3 rows of 3 | 9 sq units |
| Rectangle (5 × 3) | 5 rows of 3 | 15 sq units |
| Rectangle (4 × 4) | 4 rows of 4 (it is a square!) | 16 sq units |
'When using a grid, count ALL the FULL squares. If there are half-squares, two halves make one whole.'
6. Area of Common Shapes
Square:
Formula: Area = Side × Side (Side²)
Example: Find area of a square with side = 6 cm.
Area = 6 × 6 = 36 sq cm (36 cm²)
| Side | Area |
|---|---|
| 2 cm | 4 sq cm |
| 5 m | 25 sq m |
| 10 cm | 100 sq cm |
Rectangle:
Formula: Area = Length × Breadth
Example: Find area of a rectangle with L = 8 cm, B = 5 cm.
Area = 8 × 5 = 40 sq cm (40 cm²)
| Length | Breadth | Area |
|---|---|---|
| 7 cm | 3 cm | 21 sq cm |
| 12 m | 8 m | 96 sq m |
| 15 cm | 10 cm | 150 sq cm |
7. Perimeter vs Area — The Difference
| Perimeter | Area |
|---|---|
| Distance AROUND the shape | Space INSIDE the shape |
| Measure with a ruler (length) | Count squares (square units) |
| Unit: cm, m, km | Unit: cm², m², km² |
| Formula: Sum of all sides | Formula: Length × Width |
'Two shapes can have the SAME perimeter but DIFFERENT areas. A 5×3 rectangle has perimeter 16 and area 15, while a 6×2 rectangle also has perimeter 16 but area only 12!'
8. Real-Life Area Problems
Problem 1 — Carpet:
A room is 6 m long and 4 m wide. How much carpet is needed to cover the floor?
Solution: Area = 6 × 4 = 24 sq m of carpet.
Problem 2 — Garden:
A rectangular garden is 15 m long and 10 m wide. A 1 m wide path surrounds it. What is the area of the path?
Solution: Outer rectangle = 17 m × 12 m = 204 sq m. Inner garden = 15 × 10 = 150 sq m. Area of path = 204 - 150 = 54 sq m.
Problem 3 — Wall:
A wall is 5 m long and 3 m high. A door of 2 m × 1 m is in the wall. What area of the wall needs painting?
Solution: Wall area = 5 × 3 = 15 sq m. Door area = 2 × 1 = 2 sq m. Area to paint = 15 - 2 = 13 sq m.
9. Common Mistakes
- Confusing perimeter and area: 'Perimeter is the BOUNDARY (outside). Area is the SPACE (inside). They are DIFFERENT!'
- Wrong units: 'Perimeter is in cm (length). Area is in cm² (square units). NEVER say "the area is 24 cm" — it is 24 cm²!'
- Square vs rectangle formula: 'A square is a special rectangle where ALL sides are equal. Area = Side × Side works for squares. Area = Length × Breadth works for all rectangles (including squares).'
- Not using the same unit: 'If length is in m and breadth in cm, CONVERT first. 2 m and 50 cm = 200 cm and 50 cm. Or 2 m and 0.5 m.'
10. Key Facts to Remember
- 'Perimeter of square = 4 × Side. Area of square = Side × Side.'
- 'Perimeter of rectangle = 2 × (L + B). Area of rectangle = L × B.'
- 'Area is measured in SQUARE UNITS — cm², m², km².'
- 'Shapes with the SAME perimeter can have DIFFERENT areas.'
- 'To find a MISSING side when perimeter is known: subtract the known sides from the perimeter.'
11. Self-Test
Q1: Find the perimeter of a square with side 9 cm.
Q2: Find the area of a rectangle with L = 12 cm and B = 7 cm.
Q3: A rectangle has perimeter 30 cm. Its length is 10 cm. What is its breadth?
Q4: Find the area of a square with side 15 m.
Q5: Which has more area — a 6 cm × 6 cm square or a 7 cm × 5 cm rectangle?
Q6: A rectangular field is 40 m long and 25 m wide. Find: (a) How much wire is needed to fence it? (b) How much land does it cover?
Q7: A triangle has sides 8 cm, 9 cm, and 10 cm. Find its perimeter.
Answers:
A1: 4 × 9 = 36 cm A2: 12 × 7 = 84 cm² A3: P = 2 × (L + B). 30 = 2 × (10 + B). 15 = 10 + B. B = 5 cm. A4: 15 × 15 = 225 m² A5: Square: 6 × 6 = 36 cm². Rectangle: 7 × 5 = 35 cm². Square has more area. A6: (a) Perimeter = 2 × (40 + 25) = 130 m of wire (b) Area = 40 × 25 = 1000 m² A7: 8 + 9 + 10 = 27 cm
