Surface Areas and Volumes – Class 10 Maths Chapter 12: Formulas & Combinations
Class 10 Surface Areas & Volumes: combined solids, conversion of solids, frustum of cone formulas for CBSE & ICSE boards.
Chapter 12: Surface Areas and Volumes
12.1 Formula Reference Table
| Solid | CSA / LSA | TSA | Volume |
|---|---|---|---|
| Cuboid | 2h(l+b) | 2(lb+bh+hl) | l×b×h |
| Cube (side a) | 4a² | 6a² | a³ |
| Cylinder | 2πrh | 2πr(r+h) | πr²h |
| Cone | πrl | πr(r+l) | ⅓πr²h |
| Sphere | 4πr² | ⁴⁄₃πr³ | |
| Hemisphere | 2πr² | 3πr² | ⅔πr³ |
Where l = slant height of cone = √(r² + h²)
12.2 Combination of Solids
Many real-world objects are combinations of basic solids. For example, a tent is a cylinder topped by a cone.
• Surface area of combined solid = Sum of visible curved surface areas (don't count the surfaces where solids are joined).
• Volume of combined solid = Sum of individual volumes.
12.3 Conversion of Solids
When a solid is melted and recast into another shape, the volume remains the same.
Example: A metallic sphere of radius 4.2 cm is melted and recast into a cylinder of radius 6 cm. Find the height of the cylinder.
Volume of sphere = Volume of cylinder
⁴⁄₃ × π × (4.2)³ = π × (6)² × h
⁴⁄₃ × 74.088 = 36h → h = 98.784/36 = 2.744 cm
12.4 Frustum of a Cone
When a cone is cut by a plane parallel to its base, the portion between the base and the cut is called a frustum.
Slant height: l = √[h² + (R − r)²]
CSA: π(R + r)l
TSA: π(R + r)l + πR² + πr²
Volume: ⅓πh(R² + r² + Rr)
Indexed Topics & Examination Keywords
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