Perimeter and Area
1. Perimeter — The Distance Around
The PERIMETER of a closed shape is the TOTAL LENGTH of its boundary.
'Think of perimeter as WALKING around the edge of a shape. The distance you walk is the perimeter.'
Perimeter of a Square
A square has FOUR equal sides.
Perimeter = 4 × side
| Side | Perimeter (4 × side) |
|---|---|
| 5 cm | 20 cm |
| 12 m | 48 m |
| 8.5 cm | 34 cm |
| 25 mm | 100 mm |
Perimeter of a Rectangle
A rectangle has TWO pairs of equal sides (length and breadth).
Perimeter = 2 × (length + breadth) = 2(l + b)
Example: length = 8 cm, breadth = 5 cm P = 2 × (8 + 5) = 2 × 13 = 26 cm
| Length | Breadth | Perimeter 2(l + b) |
|---|---|---|
| 10 cm | 6 cm | 32 cm |
| 15 m | 12 m | 54 m |
| 7.5 cm | 4.5 cm | 24 cm |
'To find the perimeter of a rectangle, add the length and breadth first, then DOUBLE the result. Do NOT double each side separately and then add.'
Perimeter of a Triangle
Perimeter = side₁ + side₂ + side₃
| Triangle | Sides | Perimeter |
|---|---|---|
| Equilateral (all equal) | 6 cm each | 18 cm |
| Isosceles (2 equal) | 5 cm, 5 cm, 8 cm | 18 cm |
| Scalene (all different) | 4 cm, 7 cm, 9 cm | 20 cm |
Finding a Missing Side
If perimeter and other sides are given, subtract the known sides from the perimeter.
A rectangle has perimeter 30 cm and length 10 cm. Find breadth.
30 = 2 × (10 + b) 15 = 10 + b b = 5 cm
'Working BACKWARDS from the perimeter to find a missing side is an important skill. Use the INVERSE operation.'
2. Area — The Space Inside
The AREA of a shape is the TOTAL SPACE enclosed within its boundary.
'Area is measured in SQUARE units. Imagine covering a table with square tiles — the number of tiles is the area.'
Area of a Square
Area = side × side = s²
| Side | Area |
|---|---|
| 4 cm | 16 cm² |
| 10 m | 100 m² |
| 6.5 cm | 42.25 cm² |
Area of a Rectangle
Area = length × breadth = l × b
Example: length = 8 cm, breadth = 5 cm A = 8 × 5 = 40 cm²
| Length | Breadth | Area |
|---|---|---|
| 12 cm | 7 cm | 84 cm² |
| 15 m | 10 m | 150 m² |
| 6.5 cm | 4 cm | 26 cm² |
'Area is written as SQUARE units — cm², m², km². The small ² means 'squared' — it is NOT the same as centimetres multiplied by 2.'
Area of Irregular Shapes
Count the number of FULL squares inside the shape. If more than half a square is covered, count it as ONE. If less than half, IGNORE it.
'For irregular shapes drawn on a grid, this counting method gives a REASONABLE estimate of the area.'
3. Difference Between Perimeter and Area
| Perimeter | Area |
|---|---|
| Distance AROUND the shape | Space INSIDE the shape |
| One-dimensional (length) | Two-dimensional (length × width) |
| Measured in m, cm, mm | Measured in m², cm², mm² |
| Formula: add all sides | Formula: multiply sides |
| Example: Fencing a garden | Example: Carpeting a room |
'Perimeter is a LENGTH. Area is a SURFACE. They are DIFFERENT concepts — a small shape can have a large perimeter, and a large shape can have a small area.'
4. Word Problems
Example 1: Fencing
A rectangular garden is 25 m long and 15 m wide. What length of wire is needed to fence it?
Perimeter = 2 × (25 + 15) = 2 × 40 = 80 m.
Wire needed = 80 m.
Example 2: Tiling
A room is 6 m long and 4 m wide. How many square tiles of side 1 m are needed?
Area of room = 6 × 4 = 24 m². Area of one tile = 1 × 1 = 1 m². Number of tiles = 24 ÷ 1 = 24 tiles.
Example 3: Cost Calculation
Find the cost of carpeting a rectangular floor 8 m by 5 m at ₹120 per square metre.
Area = 8 × 5 = 40 m². Cost = 40 × 120 = ₹4,800.
| Problem | Given | Find | Operation | Answer |
|---|---|---|---|---|
| Fence a square field of side 30 m | Side = 30 m | Perimeter | 4 × 30 | 120 m |
| Tile a floor 10 m × 8 m with 1 m² tiles | l = 10 m, b = 8 m | Number of tiles | (10 × 8) ÷ 1 | 80 tiles |
| Paint a wall 12 m × 3 m at ₹50/m² | l = 12 m, b = 3 m, rate = ₹50/m² | Cost of painting | 12 × 3 × 50 | ₹1,800 |
'Read the problem carefully. Is it asking for FENCING (perimeter) or CARPETING (area)? Fencing runs around the boundary. Carpeting covers the surface.'
5. Application — Cost and Rate Problems
Formula
Total Cost = Area × Rate per unit area
Example
A rectangular plot is 50 m by 30 m. Find the cost of levelling it at ₹15 per m².
Area = 50 × 30 = 1500 m². Cost = 1500 × 15 = ₹22,500.
Key Facts to Remember
- Perimeter is a LENGTH — measured in m, cm, mm.
- Area is a SURFACE — measured in square units (m², cm², mm²).
- 'A square is a special rectangle. So the rectangle formula works for squares too (l and b would be equal).'
- Two shapes can have the same perimeter but DIFFERENT areas, and vice versa.
- Always write the CORRECT unit (m or m²) in the answer.
Common Mistakes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Confusing perimeter and area formulas | P = l × b is WRONG — that is area | P = 2(l + b) for rectangle |
| Forgetting to double in rectangle perimeter | P = l + b gives only HALF the distance | P = 2 × (l + b) |
| Writing cm instead of cm² for area | Area is two-dimensional | Area is in square units: cm² |
| Adding all sides for rectangle when formula works | You can add all 4 sides, but 2(l+b) is faster | Use the formula to save time |
Exam Focus (ICSE Class 5)
| Topic | Marks (Typical) | Question Type |
|---|---|---|
| Perimeter of square and rectangle | 3-4 marks | Direct computation and missing side |
| Area of square and rectangle | 3-4 marks | Direct computation |
| Word problems — fencing/carpeting | 4-5 marks | Real-life application with cost |
| Find missing side given perimeter/area | 3 marks | Inverse operations |
| Compare perimeter and area | 2 marks | True/False or fill in blanks |
Self-Test: 5 Questions
Q1. Find the perimeter of a rectangle with length 14 cm and breadth 9 cm.
Q2. A square has perimeter 48 m. What is the length of each side?
Q3. A rectangular hall is 12 m long and 8 m wide. Find the cost of tiling it at ₹180 per m².
Q4. A triangle has sides 9 cm, 12 cm, and 15 cm. Find its perimeter.
Q5. The area of a rectangle is 96 cm². If its length is 12 cm, find its breadth and perimeter.
Answers
A1. P = 2 × (14 + 9) = 2 × 23 = 46 cm.
A2. Side = Perimeter ÷ 4 = 48 ÷ 4 = 12 m.
A3. Area = 12 × 8 = 96 m². Cost = 96 × 180 = ₹17,280.
A4. P = 9 + 12 + 15 = 36 cm.
A5. Breadth = Area ÷ Length = 96 ÷ 12 = 8 cm. Perimeter = 2 × (12 + 8) = 2 × 20 = 40 cm.
