Surface Areas and Volumes — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Surface Areas and Volumes (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Surface Areas and Volumes (CBSE & HBSE)
Surface Areas and Volumes is highly scoring with direct formula application. CBSE tests frustum and combination solids. HBSE focuses on cube, cylinder, cone, sphere, and their combinations.
Surface Areas of Solid Figures
Formulas for Surface Areas
| Solid | Curved/Lateral SA | Total SA |
|---|---|---|
| Cube (side a) | 4a² | 6a² |
| Cuboid (l,b,h) | 2(l+b)h | 2(lb+bh+hl) |
| Cylinder (r,h) | 2πrh | 2πr(r+h) |
| Cone (r,l,h) | πrl | πr(r+l) |
| Sphere (r) | — | 4πr² |
| Hemisphere (r) | 2πr² | 3πr² |
Where l = slant height of cone = √(r²+h²)
When to Use Which Surface Area?
- Total SA: When the entire object is being painted/wrapped (includes base)
- Curved SA: When only the curved part is considered (e.g., a sheet wrapped around, without top/bottom)
- Lateral SA: Same as Curved SA for most problems
Combination Solids — Surface Area
Rule: For a combined solid (e.g., cone on top of cylinder), the total SA is the sum of individual curved SAs, but DO NOT include the surfaces that are 'hidden' (joined).
Example: Toy = Hemisphere on top of cone
- SA = Curved SA of cone + Curved SA of hemisphere
- Do NOT count the circular base (where they join)
CBSE Tip: In combination solid problems, first identify which surfaces are visible/exposed. Only count those in Total SA.
Volumes of Solid Figures
Formulas for Volumes
| Solid | Volume |
|---|---|
| Cube (a) | a³ |
| Cuboid (l,b,h) | l×b×h |
| Cylinder (r,h) | πr²h |
| Cone (r,h) | (1/3)πr²h |
| Sphere (r) | (4/3)πr³ |
| Hemisphere (r) | (2/3)πr³ |
| Frustum (R,r,h) | (π h/3)(R²+r²+Rr) |
Volume Relationships
- Cylinder = 3 × Cone (same r and h)
- Sphere = (4/3) × Cylinder with h=2r and same r → (4/3)πr³
- Hemisphere = (1/2) × Sphere
Melting and Recasting Problems
Conservation of Volume: When one solid is melted and recast as another, volume is conserved.
Volume of original = Volume of new shape
Example: A sphere radius 6 cm is melted into small spheres radius 2 cm. Find number.
- Volume of big sphere = (4/3)π(6)³ = 288π
- Volume of small sphere = (4/3)π(2)³ = (32/3)π
- Number = 288π ÷ (32π/3) = 288×3/32 = 27
Frustum of a Cone
Frustum = cone with top portion cut parallel to base.
| Property | Formula |
|---|---|
| Slant height | l = √[h²+(R-r)²] |
| Volume | (πh/3)(R²+r²+Rr) |
| CSA | π(R+r)l |
| TSA | π(R+r)l + π(R²+r²) |
CBSE Exam: Frustum questions are 4-5 marks. Know the derivation logic and all three formulas (V, CSA, TSA).
Combination Solids and Conversions
Combination Solid Problems
Many objects are combinations of basic solids. Find total volume by adding volumes.
Common Combinations:
| Object | Made Of |
|---|---|
| Pencil | Cylinder + Cone |
| Toy | Cylinder + Hemisphere |
| Capsule | Cylinder + 2 Hemispheres |
| Building with dome | Cylinder + Hemisphere |
| Funnel | Cone + Cylinder |
Approach for Combination Problems
- Identify each component solid
- Find dimensions of each
- Volume: Add all volumes
- SA: Add visible surfaces only (excluding joined faces)
Conversion: Solid to Liquid
Volume of solid = Volume of liquid displaced Use this for: object dropped in tank — rise in water level
Rise in level (h) = Volume of object / Base area of tank
Important: Channel/Path Problems
Water flows through pipe of radius r at speed v cm/s. Volume per second = πr² × v
Example: Pipe radius 3 cm, speed 5 m/s fills tank 10 m × 6 m. Find time.
- Volume per second = π × 0.03² × 500 = π × 0.45 cm³... use consistent units
HBSE Tip: Always convert all dimensions to same unit before substituting in formulas. 1 m = 100 cm, 1 m³ = 10⁶ cm³
Frequently asked questions
Are these Surface Areas and Volumes notes free?
Yes — the Surface Areas and Volumes notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Surface Areas and Volumes notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Surface Areas and Volumes chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Surface Areas and Volumes.