Chapter 16 of 18

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Mensuration - Surface Area and Volume

Master surface area and volume for ICSE Class 9. Learn formulas and applications for cube, cuboid, cylinder, cone, sphere, and hemisphere with solved examples.

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Mensuration - Surface Area and Volume

Introduction

Surface area and volume calculations are crucial for understanding three-dimensional objects. ICSE Class 9 covers cuboids, cubes, cylinders, cones, spheres, and hemispheres.

Cube and Cuboid

Cuboid

Total Surface Area (TSA) = 2(lb + bh + lh)
Lateral Surface Area (LSA) = 2h(l + b)
Volume = l × b × h
Diagonal = √(l² + b² + h²)

Cube

TSA = 6a²
LSA = 4a²
Volume = a³
Diagonal = a√3

<ICSEExample title="Cuboid Volume and Surface Area"> Find the volume, TSA, and diagonal of a cuboid with dimensions 10 cm, 8 cm, and 6 cm. <Solution> Volume = l × b × h = 10 × 8 × 6 = 480 cm³ TSA = 2(10×8 + 8×6 + 10×6) = 2(80 + 48 + 60) = 2×188 = 376 cm² Diagonal = √(100 + 64 + 36) = √200 = 10√2 cm </Solution> </ICSEExample>

Cylinder

For a cylinder with base radius r and height h:

Curved Surface Area (CSA) = 2πrh
Total Surface Area (TSA) = 2πr(r + h)
Volume = πr²h

<ICSEExample title="Cylinder Calculations"> Find the CSA, TSA, and volume of a cylinder with radius 7 cm and height 12 cm. (Use π = 22/7) <Solution> CSA = 2πrh = 2 × 22/7 × 7 × 12 = 528 cm² TSA = 2πr(r + h) = 2 × 22/7 × 7(7 + 12) = 44 × 19 = 836 cm² Volume = πr²h = 22/7 × 7 × 7 × 12 = 1848 cm³ </Solution> </ICSEExample>

Cone

For a cone with base radius r, height h, and slant height l:

l = √(r² + h²)
CSA = πrl
TSA = πr(r + l)
Volume = 1/3 × πr²h

<ICSEExample title="Cone Calculations"> Find the slant height, CSA, TSA, and volume of a cone with radius 6 cm and height 8 cm. <Solution> l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm CSA = πrl = 22/7 × 6 × 10 = 188.57 cm² TSA = πr(r + l) = 22/7 × 6(6 + 10) = 22/7 × 96 = 301.71 cm² Volume = 1/3 × πr²h = 1/3 × 22/7 × 36 × 8 = 301.71 cm³ </Solution> </ICSEExample>

Sphere and Hemisphere

Sphere

Surface Area = 4πr²
Volume = 4/3 × πr³

Hemisphere

Curved Surface Area = 2πr²
Total Surface Area = 3πr² (includes the circular base)
Volume = 2/3 × πr³

<ICSEExample title="Sphere and Hemisphere"> Find the surface area and volume of a sphere with radius 10.5 cm. Also find the TSA and volume of the corresponding hemisphere. <Solution> Sphere: Surface area = 4πr² = 4 × 22/7 × 10.5 × 10.5 = 1386 cm² Volume = 4/3 × πr³ = 4/3 × 22/7 × 10.5³ = 4851 cm³

Hemisphere: TSA = 3πr² = 3 × 22/7 × 10.5 × 10.5 = 1039.5 cm² Volume = 2/3 × πr³ = 2/3 × 22/7 × 10.5³ = 2425.5 cm³ </Solution> </ICSEExample>

Summary of Formulas

Shape	CSA/LSA	TSA	Volume
Cube	4a²	6a²	a³
Cuboid	2h(l+b)	2(lb+bh+lh)	lbh
Cylinder	2πrh	2πr(r+h)	πr²h
Cone	πrl	πr(r+l)	(1/3)πr²h
Sphere	-	4πr²	(4/3)πr³
Hemisphere	2πr²	3πr²	(2/3)πr³

Common Mistakes With Fixes

Mistake	Correction
Forgetting the 1/3 factor in cone volume	Volume of cone = 1/3 × πr²h (not πr²h)
Using radius instead of slant height for cone CSA	CSA of cone = πrl, not πrh
Confusing curved and total surface area	CSA excludes bases; TSA includes all surfaces
Mixing sphere and hemisphere formulas	Hemisphere has half the volume of sphere

ICSE Exam Focus

Topic	Marks (approx.)	Frequency
Volume and surface area of cylinder	4-5 marks	Very common
Cone and sphere calculations	4-5 marks	Common
Combined solids (mixed shapes)	5-6 marks	Frequently asked
Finding missing dimensions	4 marks	Common

Self-Test

Q1: Find the volume and TSA of a cube with side 8 cm.

Q2: A cylinder has radius 5 cm and height 14 cm. Find its CSA and volume.

Q3: Find the slant height and TSA of a cone with radius 9 cm and height 12 cm.

Q4: The surface area of a sphere is 616 cm². Find its radius and volume.

Q5: A cuboid measures 12 cm × 10 cm × 8 cm. Find its diagonal length.

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