What is Mensuration?
Mensuration is the branch of mathematics that deals with measurement of geometric figures — their perimeter, area, volume, etc.
At Class 6 level, we focus on:
- Perimeter — length of the boundary (1D measurement)
- Area — surface enclosed within the boundary (2D measurement)
Perimeter
Perimeter = total length of the boundary of a closed figure.
Unit: cm, m, km (same as length)
General rule: Add up all the sides.
Perimeter of Rectangle
Perimeter = 2 × (l + b)
Example: Rectangle with l = 8 cm, b = 5 cm
- Perimeter = 2 × (8 + 5) = 2 × 13 = 26 cm
Perimeter of Square
All 4 sides are equal.
Perimeter = 4 × side = 4s
Example: Square with side = 7 cm
- Perimeter = 4 × 7 = 28 cm
Perimeter of Triangle
Perimeter = sum of all three sides = a + b + c
Example: Triangle with sides 5 cm, 7 cm, 9 cm
- Perimeter = 5 + 7 + 9 = 21 cm
Perimeter of Equilateral Triangle
All 3 sides equal.
Perimeter = 3 × side
Perimeter of Regular Polygon
Perimeter = n × side (where n = number of sides)
Example: Regular hexagon with side 4 cm
- Perimeter = 6 × 4 = 24 cm
Area
Area = amount of surface enclosed within a boundary.
Unit: cm² (square centimetres), m² (square metres)
Intuition: Area counts how many unit squares fit inside the shape.
Area of Rectangle
Area = length × breadth = l × b
Example: Rectangle with l = 6 cm, b = 4 cm
- Area = 6 × 4 = 24 cm²
Area of Square
Area = side × side = s²
Example: Square with side = 5 cm
- Area = 5 × 5 = 25 cm²
Counting Squares Method
For irregular shapes on grid paper:
- Count full squares → each = 1 sq. unit
- Count half squares (more than half) → each = 1 sq. unit
- Ignore squares less than half
Formulas Summary
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle | 2(l + b) | l × b |
| Square | 4s | s² |
| Triangle | a + b + c | (not in Class 6) |
| Equilateral triangle | 3s | — |
| Regular polygon (n sides) | n × s | — |
Relationship Between Perimeter and Area
Two shapes can have the same perimeter but different areas (or same area but different perimeters).
Example: Both have same perimeter of 20 cm:
- Rectangle 6×4: Area = 24 cm²
- Rectangle 7×3: Area = 21 cm²
- Square 5×5: Area = 25 cm² ← largest area for same perimeter
A square has the maximum area among all rectangles with the same perimeter.
Key Vocabulary
| Term | Meaning |
|---|---|
| Perimeter | Total boundary length |
| Area | Total surface enclosed |
| Square unit | Unit of area (cm², m²) |
| Length | Longer side of a rectangle |
| Breadth | Shorter side of a rectangle |
| Side | Length of one edge of a square |
Exam Tips
- Perimeter is 1D (length units); Area is 2D (square units) — never mix up.
- For rectangle: if given perimeter, find length: l = (P/2) − b.
- Square has maximum area for a given perimeter — a comparison question.
- Area by counting squares: full squares count as 1, half squares as ½ (or count those more than half as 1, less as 0 — follow NCERT convention).
- Area of a border: area of outer − area of inner rectangle.
- Units: 1 m = 100 cm, so 1 m² = 10,000 cm² — unit conversion is a common trap.