What is the formula for the surface area of a sphere?

Surface area of a sphere = 4 × π × r².

How to calculate the volume of a cone?

Volume of a cone = (1/3) × π × r² × h.

Are these notes useful for competitive exams?

Yes, these notes help NEET, SSC, Railway, and Police Bharti exam preparation as well as Class 10 board exams.

Surface Areas & Volumes — Class 10 Maths Notes

📘 Introduction

This chapter covers Surface Areas and Volumes of cube, cuboid, sphere, hemisphere, cylinder, cone, and combination solids. Learn all formulas, solved examples, and practice questions. Essential for NEET, SSC, Railway, Police Bharti and Class 10 exams.

🔹 Key Formulas

Cube Surface Area: 6 × a²
Cuboid Surface Area: 2(lb + bh + hl)
Sphere Surface Area: 4 × π × r²
Hemisphere Surface Area: 3 × π × r²
Cylinder Surface Area: 2πr(h+r)
Cone Surface Area: πr(l+r)
Volume of Cube: a³
Volume of Cuboid: l × b × h
Volume of Sphere: 4/3 × π × r³
Volume of Hemisphere: 2/3 × π × r³
Volume of Cylinder: π × r² × h
Volume of Cone: 1/3 × π × r² × h

📊 Chapter Summary Table

Solid	Surface Area	Volume
Cube	6a²	a³
Cuboid	2(lb+bh+hl)	l×b×h
Sphere	4πr²	4/3πr³
Hemisphere	3πr²	2/3πr³
Cylinder	2πr(h+r)	πr²h
Cone	πr(l+r)	1/3πr²h

🧮 Practice Questions

Find the surface area of a sphere with radius 7 cm.
Calculate the volume of a cone with radius 5 cm and height 12 cm.
Surface area of a cylinder with radius 8 cm and height 10 cm?
Volume of a hemisphere with radius 6 cm?
Find the total surface area of a cuboid with l=10 cm, b=6 cm, h=4 cm.
Volume of a cube with edge 5 cm?

1. Cube & Cuboid

Cuboid with length l, breadth b and height h:

Total Surface Area (TSA) = 2(lb + bh + hl) ; Volume = l × b × h

Cube with side a:

TSA = 6a² ; Volume = a³

Visual tip: TSA equals sum of areas of all faces. Practice drawing nets to visualise.

2. Cylinder

Right circular cylinder with radius r and height h:

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

Note: CSA comes from 'unrolling' the curved surface into a rectangle of height h and width circumference 2πr.

3. Cone & Frustum

Right circular cone with base radius r, slant height l and vertical height h:

CSA = πrl ; TSA = πr(l + r) ; Volume = (1/3)πr²h

Frustum (truncated cone) with radii r₁, r₂ and slant height l:

CSA = πl(r₁ + r₂) ; Volume = (1/3)πh(r₁² + r₂² + r₁r₂)

Derive frustum formula by subtracting smaller cone from larger cone when needed.

4. Sphere & Hemisphere

Sphere with radius r:

Surface Area = 4πr² ; Volume = (4/3)πr³

Hemisphere (solid half-sphere):

Curved Surface Area = 2πr² ; TSA (including base) = 3πr² ; Volume = (2/3)πr³

5. Nets & Visual Understanding

Drawing nets helps to visualise which faces contribute to surface area problems. Common nets:

Cuboid: 6 rectangles
Cube: 6 equal squares
Cylinder: rectangle (2πr × h) + 2 circles
Cone: sector of circle (for curved part) + circle (base)

Practice sketching nets — many exam questions are net-based (shaded surfaces, paint coverage).

6. Units & Conversion

Surface area units are square units (m², cm²). Volume units are cubic units (m³, cm³). Always convert to same units before calculations — e.g., convert cm to m or vice versa.

Tip: To convert cm³ to m³ divide by 1,000,000 (10⁶). To convert m³ to cm³ multiply by 10⁶.

7. Solved Examples

Example 1. Find volume and TSA of cylinder with r = 7 cm and h = 10 cm.

Volume = π×7²×10 = 490π cm³. TSA = 2π×7(10 + 7) = 14π×17 = 238π cm².

Example 2. A cone has base radius 3 cm and slant height 5 cm. Find CSA and volume if vertical height is 4 cm.

CSA = π×3×5 = 15π cm². Volume = (1/3)π×3²×4 = 12π cm³.

Example 3. A sphere has volume 288π cm³. Find its radius and surface area.

(4/3)πr³ = 288π ⇒ r³ = 216 ⇒ r = 6 cm. Surface Area = 4πr² = 4π×36 = 144π cm².

Example 4 (Paint problem). A cuboid tank 2 m × 1.5 m × 1 m is painted on outer surface. If 1 litre of paint covers 10 m², how many litres are needed? TSA = 2(lb + bh + hl) = 2(3 + 1.5 + 2) = 13 m² ⇒ paint = 13/10 = 1.3 litres (round as instructed).

8. Practice Questions

Find TSA and volume of a cube whose side is 5 cm.
Curved surface area of a right circular cylinder is 66π cm² and its height is 11 cm. Find radius and volume.
A cone and a hemisphere have same base radius 7 cm. If cone height is 24 cm, find which solid has greater volume and by how much.
Frustum: A cone of height 15 cm and base radius 9 cm is cut by a plane parallel to base at distance 6 cm from vertex. Find volume of frustum formed.

9. Exam Tips & Shortcuts

Remember common factors of π — cancel early when possible.
For paint/metal sheet problems, pay attention to whether base is included (TSA) or not (CSA).
Sketch nets for surface area questions — they reduce mistakes.
Use volume subtraction for frustum and hollow solids (subtract inner from outer).