Area — Class 8 Mathematics (Ganita Prakash)

"Area is how the world measures itself — from the smallest leaf to the vastest ocean."

1. About the Chapter

'Area' is the closing chapter of Class 8 maths. It consolidates and extends your knowledge of:

• Area of plane figures (triangle, quadrilaterals, circle)
• Composite figures (combination shapes)
• Surface area of 3D solids (cube, cuboid, cylinder)
• Volume of 3D shapes

These formulas are used everywhere — architecture, engineering, packaging, agriculture, science.

2. Area Formulas — Plane Figures

Triangle

Area = (1/2) × base × height

For a right triangle: A = (1/2) × leg₁ × leg₂

Heron's formula (when 3 sides a, b, c known):

• Semi-perimeter s = (a + b + c)/2
• Area = √(s(s−a)(s−b)(s−c))

Rectangle

Area = length × breadth = L × B

Square

Area = side² = a²

Parallelogram

Area = base × height (where height is perpendicular distance between parallel sides)

Rhombus

Area = (1/2) × d₁ × d₂ (half the product of diagonals)

Alternatively: Area = base × height

Trapezium

Area = (1/2) × (sum of parallel sides) × height = (1/2)(a + b)h, where a, b are parallel sides and h is perpendicular distance

Kite

Area = (1/2) × d₁ × d₂ (same as rhombus)

Circle

• Area = πr²
• Circumference = 2πr
• Use π = 22/7 or 3.14

Sector (pie-slice of circle)

• Area = (θ/360°) × πr²
• where θ = angle of the sector

3. Composite Figures

Strategy

Composite figures (shapes made by combining simpler shapes) are tackled by:

1. DIVIDE into simple shapes
2. Calculate area of each part
3. ADD the parts (or SUBTRACT for cut-out shapes)

Example

A swimming pool is rectangular (10m × 6m) with a semicircular extension on one end (radius 3m).

• Rectangle area = 10 × 6 = 60 m²
• Semicircle area = (1/2)π(3)² = 4.5π ≈ 14.14 m²
• Total area = 60 + 14.14 = 74.14 m²

Subtraction Example

A square of side 10 cm has a circle of radius 3 cm cut out from the centre.

• Square area = 100 cm²
• Circle area = π(3)² = 9π ≈ 28.27 cm²
• Remaining = 100 − 28.27 = 71.73 cm²

4. Surface Area of 3D Solids

Cube

• Surface area = 6a² (6 faces, each a²)
• Volume = a³

Cuboid (Rectangular Box)

• Surface area = 2(LB + BH + HL)
• Volume = L × B × H

Where L = length, B = breadth, H = height.

Lateral Surface Area

Lateral SA (cuboid) = 2(L + B) × H — surfaces excluding top and bottom.

Cylinder

• Curved surface area (CSA) = 2πrh
• Total surface area (TSA) = 2πr(r + h) (add two circular ends)
• Volume = πr²h

Where r = radius, h = height.

5. Worked Examples

Example 1: Triangle Area

Triangle with base 8 cm and height 5 cm. Find area.

• A = (1/2) × 8 × 5 = 20 cm²

Example 2: Heron's Formula

Triangle with sides 5, 12, 13 cm.

• s = (5+12+13)/2 = 15
• Area = √(15 × 10 × 3 × 2) = √900 = 30 cm²

(Note: this is a right triangle with legs 5, 12. Area = (1/2)(5)(12) = 30 cm² ✓)

Example 3: Parallelogram

Base 12 cm, height 7 cm.

• A = 12 × 7 = 84 cm²

Example 4: Rhombus

Diagonals 16 cm and 10 cm.

• A = (1/2)(16)(10) = 80 cm²

Example 5: Trapezium

Parallel sides 14 cm and 10 cm; height 6 cm.

• A = (1/2)(14 + 10)(6) = (1/2)(24)(6) = 72 cm²

Example 6: Circle

Radius 14 cm (use π = 22/7).

• A = π(14)² = (22/7)(196) = 616 cm²

Example 7: Composite Figure

A rectangle 12 × 8 m has a semicircular path on one short side (radius = 4 m).

• Rectangle = 12 × 8 = 96 m²
• Semicircle = (1/2)π(4)² = 8π ≈ 25.13 m²
• Total = 96 + 25.13 = 121.13 m²

Example 8: Cube Surface and Volume

Cube of side 5 cm.

• SA = 6 × 5² = 150 cm²
• V = 5³ = 125 cm³

Example 9: Cuboid

Cuboid 10 × 6 × 4 cm.

• SA = 2(10×6 + 6×4 + 4×10) = 2(60 + 24 + 40) = 2(124) = 248 cm²
• V = 10 × 6 × 4 = 240 cm³

Example 10: Cylinder

Cylinder with radius 7 cm and height 10 cm. (π = 22/7)

• CSA = 2 × 22/7 × 7 × 10 = 440 cm²
• TSA = 2 × 22/7 × 7 × (7 + 10) = 748 cm²
• V = 22/7 × 7² × 10 = 1540 cm³

6. Real-World Applications

Architecture

• Floor area of rooms (rectangular formula)
• Ceiling area for painting (square/rectangle/composite)
• Garden area (composite figures)

Construction

• Volume of concrete for foundation
• Surface area of walls for tiles
• Roofing area (triangles, trapeziums)

Agriculture

• Plot area (rectangle/triangle for irregular plots)
• Crop yield per area
• Fertiliser distribution

Packaging

• Box surface area (cardboard required)
• Cylinder volume (bottles, cans)
• Sphere surface (balls)

Industrial

• Material requirements
• Heat exchange surfaces
• Tank capacities

7. Common Mistakes

1. Wrong height in parallelogram

   • Height is PERPENDICULAR distance, NOT slant side

2. Diagonal vs side in rhombus

   • Rhombus area = (1/2) × d₁ × d₂ (NOT side²)

3. Forgetting (1/2) in triangle

   • Triangle = (1/2) × b × h (the 1/2 is essential)

4. Surface area of cylinder formulas

   • CSA = 2πrh (no r at end)
   • TSA = 2πr(r + h) (or 2πrh + 2πr²)

5. Mixing units

   • Don't mix cm and m
   • Convert all to same unit first

6. Wrong π value

   • Use 22/7 if radius/dia is multiple of 7
   • Use 3.14 otherwise (or as specified)

8. Tips for Mastery

Memorise All Formulas

Make a one-page summary sheet:

• Triangle: (1/2)bh
• Rectangle: LB
• Parallelogram: bh
• Rhombus: (1/2)d₁d₂
• Trapezium: (1/2)(a+b)h
• Circle: πr²
• Cube: 6a² (SA), a³ (V)
• Cuboid: 2(LB+BH+HL) (SA), LBH (V)
• Cylinder: 2πr(r+h) (TSA), πr²h (V)

Always

• Draw a diagram
• Label all dimensions
• Use consistent units
• Show all steps
• Include units in the final answer

Practice

• 5 problems of each shape
• Mix composite figures
• Real-world word problems

9. Surface Area vs Volume

Different Concepts

• Surface area: how much MATERIAL covers the outside (in square units)
• Volume: how much SPACE fills the inside (in cubic units)

Why They Matter

• Painting a room: surface area
• Filling a tank: volume
• Wrapping a gift: surface area
• Storing items: volume

Units

• Area: m², cm², mm², km²
• Volume: m³, cm³, mm³, litres (1 L = 1000 cm³)

10. Indian Mathematical Heritage

Sulba Sutras

• Earliest precise area calculations for fire altars
• Used Pythagorean relationships for area

Bhaskara II's 'Lilavati'

• Area problems in poetic form
• Many algebraic methods for area calculations

Brahmagupta's Formula

For cyclic quadrilateral with sides a, b, c, d: Area = √((s−a)(s−b)(s−c)(s−d)) where s = (a+b+c+d)/2.

This is Heron's formula generalised to cyclic quadrilaterals.

11. Conclusion

'Area' is the chapter that closes Class 8 mathematics on a practical note. The formulas here are used in every profession that involves physical things:

• Architects calculate floor areas
• Engineers compute volumes of materials
• Farmers measure land
• Manufacturers design packaging
• Doctors compute body surface area for drug dosing

Master these formulas, practise composite figures, and remember the units. Area and volume are not abstract concepts — they are the language by which we measure the world.

In Class 9, you'll add new shapes (cones, spheres, hemispheres) and continue this geometric journey. The foundation laid here makes that journey smooth.