Mensuration — Class 6 IMO

Perimeter

The perimeter is the total distance around a shape. For a square it is 4 × side, for a rectangle 2 × (length + breadth), and for a regular polygon it is number of sides × side length.

So a regular pentagon of side 6 cm has perimeter 5 × 6 = 30 cm.

Example 1: Find the perimeter of a regular pentagon of side 6 cm.

5 × 6 = 30 cm.

Example 2: Find the perimeter of a rectangle 7 cm by 3 cm.

2 × (7 + 3) = 20 cm.

Quick recap

Perimeter = distance around a shape.
Square: 4 × side; rectangle: 2 × (l + b); regular polygon: sides × side length.

✓ Quick check

What is the perimeter of a square of side 9 cm?

4 × 9 = 36 cm.

What is the perimeter of an equilateral triangle of side 5 cm?

3 × 5 = 15 cm.

Area of Square, Rectangle & Right Triangle

What is Area?

Area is the amount of space inside a 2-dimensional shape. It is measured in square units (cm², m², in², etc.).

Why is Triangle Area Formula ½ × base × height?

A triangle is exactly half of a parallelogram (or rectangle) with the same base and height!

Rectangle area = base × height
Triangle area = ½ × base × height

Important Notes:

Base can be any side of the triangle
Height must be perpendicular (straight up/down) to the base
Height is NOT the slanted side!

TRIANGLE AS HALF A RECTANGLE:

 Rectangle: Triangle:
 ┌─────────────────┐ ╱│
 │ │ ╱ │
 │ │ ╱ │
 │ height │ ╱ │ height
 │ │ ╱ │
 │ │ ╱ │
 └─────────────────┘ ╱______┘
 ←─── base ───→ ← base →
 
 Area = b × h Area = (b × h) ÷ 2


IDENTIFYING BASE AND HEIGHT:

 Acute Triangle: Right Triangle:
 /│ /|
 / │ / |
 / │ / |
 / │h /h |
 / │ / |
 └─────┴─────┘ └─────┘
 base base
 
 Obtuse Triangle (height outside):
 
 /│
 / │ 
 / │ 
 / │ 
 / │ 
 └─────┴─────┘
 base
 (height drawn to extended base)


DIFFERENT BASE CHOICES:

 Same triangle can have different base-height pairs:
 
 Base = b₁ Base = b₂
 height = h₁ height = h₂
 
 /│
 / │ 
 / │ 
 / │h₁
 / │ 
 └──b₁─┴────┘
 
 Area = ½ × b₁ × h₁ = ½ × b₂ × h₂ (always the same!)

Example 1: Find the area of a rectangle 8 cm by 5 cm.

8 × 5 = 40 sq cm.

Example 2: Find the area of a right triangle with legs 6 cm and 4 cm.

½ × 6 × 4 = 12 sq cm.

Example 3:

Find the area of a triangle with base 8 cm and height 5 cm.

A = (1)/(2) × b × h
Area = ½ × 8 × 5
Area = 4 × 5 = 20
Answer: 20 cm²

Example 4:

A triangle has area 30 cm² and base 10 cm. Find its height.

A = (1)/(2) × b × h
30 = ½ × 10 × h
30 = 5 × h
h = 30 ÷ 5 = 6
Answer: 6 cm

Example 5:

Find the area of a right triangle with legs 6 cm and 8 cm.

In a right triangle, the legs are perpendicular
Use one leg as base, the other as height
Area = ½ × 6 × 8 = ½ × 48 = 24
Answer: 24 cm²

Quick recap

Square: side × side; rectangle: length × breadth.
Right triangle: ½ × base × height.
Triangle area = ½ × base × height
Base and height must be perpendicular
Height is NOT the slanted side
Units are square units (²)
Any side can be the base as long as you use its perpendicular height

✓ Quick check

What is the area of a square of side 7 cm?

7 × 7 = 49 sq cm.

What is the area of a right triangle with base 10 cm and height 8 cm?

½ × 10 × 8 = 40 sq cm.

Area by Grid & Word Problems

What is an Irregular Polygon?

An irregular polygon is a shape that is not a standard shape like a triangle, square, or rectangle. The sides are not all equal, and angles may vary.

Two Methods to Find Area:

Method	Description	When to Use
Decomposing	Break shape into smaller shapes (triangles, rectangles)	Shape has "indents" or can be split easily
Composing	Add extra shapes to form a larger shape, then subtract	Shape has "missing pieces"

Strategy:

Look for ways to split the shape into rectangles, triangles, and parallelograms
Find area of each part
Add all areas together

DECOMPOSING IRREGULAR POLYGON:

 L-shaped figure:
 
 ┌─────────┐
 │ │
 │ A │ ┌───┐
 │ │ │ C │
 │ │ │ │
 ├─────┐ │ │ │
 │ B │ │ │ │
 │ │ │ │ │
 └─────┴───┘ └───┘
 
 Method 1: Split into rectangles A and B
 Area = A + B
 
 Method 2: Split into rectangles A, B, and C
 Area = A + B + C


DECOMPOSING INTO TRIANGLES AND RECTANGLES:

 Irregular pentagon:
 
 /│
 / │ 
 / │ 
 / │ 
 / │ 
 └─────┴─────┘
 
 Split into:
 - Rectangle in middle
 - Triangle on top
 - Triangles on sides (or just one triangle if symmetrical)


COMPOSING (ADD THEN SUBTRACT):

 Shape with a "hole":
 
 ┌─────────────┐
 │ │
 │ ┌─────┐ │
 │ │ │ │
 │ └─────┘ │
 │ │
 └─────────────┘
 
 Area = Big rectangle - Small rectangle


COMPLEX SHAPE EXAMPLE:

 Arrow shape:
 
 ┌───┐
 │ │
 ┌───┼───┼───┐
 │ │ │ │
 └───┼───┼───┘
 │ │
 └───┘
 
 Decompose into:
 - Center rectangle
 - Left rectangle
 - Right rectangle
 - Bottom rectangle

Example 1: A figure on a grid covers 12 full unit squares. What is its area?

12 square units.

Example 2: A garden is 20 m by 15 m. How much fencing is needed?

Perimeter = 2 × (20 + 15) = 70 m.

Example 3:

Find the area of this L-shaped figure (all measurements in cm):

Top rectangle: 8 cm × 3 cm
Bottom rectangle: 5 cm × 4 cm (overlaps with top by 3 cm)

Top rectangle area = 8 × 3 = 24 cm²
Bottom rectangle width = 5 cm, height = 4 cm, area = 20 cm²
But the overlapping corner (3×3=9) is counted twice if we just add
Better: Split into non-overlapping rectangles
Rectangle A: 8 × 3 = 24
Rectangle B: 5 × 4 = 20
No overlap if split differently!
Total = 24 + 20 = 44
Answer: 44 cm²

Example 4:

Find the area of a shape made of a 6×4 rectangle with a 3×2 square attached on top.

Rectangle: 6 × 4 = 24 cm²
Square: 3 × 2 = 6 cm²
Total = 24 + 6 = 30
Answer: 30 cm²

Example 5:

A shape consists of a triangle (base 8 cm, height 5 cm) attached to a rectangle (8 cm × 4 cm). Find total area.

Triangle area = ½ × 8 × 5 = 20 cm²
Rectangle area = 8 × 4 = 32 cm²
Total = 20 + 32 = 52
Answer: 52 cm²

Quick recap

Grid method: count the unit squares inside.
Fencing → perimeter; carpet/paint → area.
Decompose = break into smaller shapes
Compose = add pieces to make larger shape, then subtract
Label all dimensions clearly
Add areas of all parts
Check for overlaps (don't count twice!)

✓ Quick check

A figure covers 9 unit squares on a grid. What is its area?

9 unit squares = 9 sq cm.

A room floor is 6 m by 5 m. How much carpet is needed?

Area = 6 × 5 = 30 sq m.