Chapter 9: Mensuration — Class 8 Maths Notes | Areas, Volumes & Surface Areas

1. Basic Concepts

Perimeter: total length around a 2D shape. Area: space enclosed by a 2D shape (in square units). Surface Area: total area of surfaces of a 3D object. Volume: space occupied by a 3D object (in cubic units).

Always include proper units: cm, m, mm², cm², m³, etc. Convert units before calculating (e.g., cm → m).

2. Area & Perimeter — Common 2D shapes

Shape	Perimeter	Area
Rectangle	`2(l + b)`	`l × b`
Square	`4a`	`a²`
Triangle	`sum of sides`	`(1/2) × base × height`
Parallelogram	`2(a + b)`	`base × height`
Trapezium	`sum of sides`	`((a + b)/2) × height`
Circle	`2πr`	`πr²`

Use π ≈ 22/7 or π ≈ 3.1416 as instructed. For composite shapes, split into basic shapes and sum areas.

3. Surface Area — Common 3D solids

Solid	Surface Area
Cuboid	`2(lw + lh + wh)`
Cube	`6a²`
Cylinder	`2πr(h + r)` (CSA = 2πrh)
Cone	`πr(l + r)` where l = slant height
Sphere	`4πr²`

CSA = curved surface area, TSA = total surface area. Remember to add areas of bases where required.

4. Volume — Common 3D solids

Solid	Volume
Cuboid	`l × w × h`
Cube	`a³`
Cylinder	`πr²h`
Cone	`(1/3)πr²h`
Sphere	`(4/3)πr³`

For frustums or composite solids, calculate volume by subtracting or adding simpler solids.

5. Solved Examples

Example 1: Area of rectangle with length 12 m and breadth 7 m.

Area = 12 × 7 = 84 m².

Example 2: Volume of cylinder with r = 5 cm, h = 10 cm.

Volume = πr²h = π × 25 × 10 = 250π cm³ ≈ 785.4 cm³ (use π≈3.1416).

Example 3: TSA of a cone with r = 3 cm and slant height l = 5 cm.

TSA = πr(l + r) = π × 3 × (5 + 3) = 24π cm² ≈ 75.40 cm².

Example 4: Surface area of a cube side 4 cm → 6×4² = 96 cm².

6. Practice Questions (with answers)

Find the area of a triangle with base 10 cm and height 6 cm.
Answer
Area = (1/2)×10×6 = 30 cm²
Find curved surface area of a cylinder with r = 7 cm and h = 14 cm.
Answer
CSA = 2πrh = 2π×7×14 = 196π ≈ 615.75 cm²
Volume of a cube with side 3 m.
Answer
V = 3³ = 27 m³
A solid sphere radius 6 cm — find volume.
Answer
V = (4/3)πr³ = (4/3)π×216 = 288π ≈ 904.78 cm³

Show steps and units in each answer. For exam problems, sketch the solid and label given dimensions before calculation.

7. Exam Tips & Tricks

Always convert units to match before computing (e.g., cm → m when required).
Draw a diagram and label known values — it reduces mistakes.
For composite shapes, decompose into simple shapes and add/subtract areas or volumes.
Remember which formulas use π and use the correct approximation as instructed.
Write units for final answers (e.g., cm² for area, cm³ for volume).