Surface Areas and Volumes — Notes
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12.1 Combination of Solids
A solid may be formed by joining or removing familiar solids such as cuboids, cylinders, cones, hemispheres and spheres. For surface area, count only the surfaces that are actually exposed.
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12.2 Conversion of Solids
When one solid is melted and recast into another, the volume is conserved, provided there is no loss of material.
Core principle
Volume before = Volume after.
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12.3 Frustum of a Cone
A frustum is obtained when a cone is cut by a plane parallel to its base. If radii are R and r and slant height is l:
CSA = π(R+r)l
Volume = (1/3)πh(R2+r2+Rr).
CSA = π(R+r)l
Volume = (1/3)πh(R2+r2+Rr).
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12.4 Important Formulae
Cylinder: CSA=2πrh, TSA=2πr(h+r), V=πr²h.
Cone: l=√(r²+h²), CSA=πrl, TSA=πr(l+r), V=⅓πr²h.
Sphere: SA=4πr², V=4/3πr³.
Hemisphere: CSA=2πr², TSA=3πr², V=2/3πr³.
Cone: l=√(r²+h²), CSA=πrl, TSA=πr(l+r), V=⅓πr²h.
Sphere: SA=4πr², V=4/3πr³.
Hemisphere: CSA=2πr², TSA=3πr², V=2/3πr³.
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12.5 Unit Conversion
For volume, remember that 1 cm³ = 1 mL and 1000 cm³ = 1 L. Convert units before applying formulas.
⚡ Quick Revision
- Use exposed surfaces only for total surface area of combinations.
- Recasting → volume conserved.
- Cone slant height l=√(r²+h²).
- Frustum volume = ⅓πh(R²+r²+Rr).
🎯 Board Exam Focus
- Separate CSA, TSA and volume clearly.
- Do not include hidden circular faces in a joined solid.
- Convert all dimensions to one unit before calculating.
🚀 Last-Minute Revision
✔ Read every formula with its conditions and symbols.
✔ Write given data first and keep units consistent.
✔ For proofs, state the theorem and show logical steps.
✔ Check signs, substitutions and final units.
❤️ Examisthan.com • Class 10 CBSE Mathematics • Chapter 12: Surface Areas and Volumes