Surface Areas and Volumes Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 12 notes: surface area and volume formulas for cuboid, cube, cylinder, cone, sphere and hemisphere, and combinations of solids.
Surface Areas and Volumes — Class 10 Maths Notes
Chapter Snapshot
This chapter is all about 3-D solids — the cuboid, cube, cylinder, cone, sphere, and hemisphere — and the solids you get by combining them (a capsule, an ice-cream cone, a circus tent, a toy). You compute their surface areas and volumes by using the standard formulas and adding or subtracting parts.
Board relevance: a Mensuration scorer, usually a 3-mark combination-surface-area and a 3–4 mark volume problem. Getting the right formula for each part and remembering which surfaces are hidden is the whole game.
Syllabus note (rationalised): the frustum of a cone and conversion of solids from one shape to another have been removed. Focus on combinations of solids.
Key Concepts & Definitions
- Curved / Lateral Surface Area (CSA/LSA) — area of only the curved or side surface.
- Total Surface Area (TSA) — CSA plus all the flat faces.
- Volume — the space a solid occupies.
- For a cone, the slant height l relates to r and h by l = √(r² + h²).
Formulas
Solid CSA / LSA TSA Volume
Cuboid (l, b, h) 2h(l + b) 2(lb + bh + hl) l·b·h
Cube (side a) 4a² 6a² a³
Cylinder (r, h) 2πrh 2πr(r + h) πr²h
Cone (r, h, slant l) πrl πr(l + r) (1/3)πr²h
Sphere (r) 4πr² 4πr² (4/3)πr³
Hemisphere (r) 2πr² 3πr² (2/3)πr³
- Sphere: CSA = TSA = 4πr² (no flat face).
- Hemisphere TSA = curved 2πr² + flat circle πr² = 3πr².
- Cone slant height: l = √(r² + h²).
- π = 22/7 or 3.14.
Combinations of Solids
Surface area of a combination: add only the exposed surfaces; the surfaces where two solids meet are hidden and not counted.
Volume of a combination: simply add (or subtract, for hollowed-out shapes) the volumes of the parts.
Common combined shapes:
- Capsule / medicine tablet = cylinder + two hemispheres.
- Ice-cream cone = cone + hemisphere (on top).
- Circus tent = cylinder (bottom) + cone (top).
- Toy top / gilli = cone + hemisphere (or two cones base-to-base).
Worked Examples
Example 1 — Cylinder: A cylinder has radius 7 cm and height 10 cm. Find its TSA and volume (π = 22/7).
TSA = 2πr(r + h) = 2(22/7)(7)(17) = 748 cm². Volume = πr²h = (22/7)(49)(10) = 1540 cm³.
Example 2 — Cone slant & CSA: A cone has r = 6 cm, h = 8 cm. Find the slant height and CSA (π = 3.14).
l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. CSA = πrl = 3.14 × 6 × 10 = 188.4 cm².
Example 3 — Combination surface area: A toy is a cone (radius 3.5 cm, slant 6 cm) mounted on a hemisphere (radius 3.5 cm). Find its total surface area (π = 22/7).
Cone CSA = πrl = (22/7)(3.5)(6) = 66 cm². Hemisphere CSA = 2πr² = 2(22/7)(12.25) = 77 cm².
The flat circle where they join is hidden, so TSA = 66 + 77 = 143 cm².
Example 4 — Combination volume: A capsule is a cylinder (radius 3 mm, length 8 mm between the flat ends) with a hemisphere at each end. Find its volume (π = 22/7).
Cylinder = πr²h = (22/7)(9)(8) = 226.29 mm³. Two hemispheres = one sphere = (4/3)πr³ = (4/3)(22/7)(27) = 113.14 mm³.
Total ≈ 339.4 mm³.
Example 5 — Cavity (subtract): A solid cylinder (radius 7 cm, height 10 cm) has a cone of the same radius and height scooped out of it. Find the remaining volume (π = 22/7).
Cylinder = πr²h = (22/7)(49)(10) = 1540 cm³. Cone = (1/3)πr²h = (1/3)(1540) = 513.33 cm³.
Remaining = 1540 − 513.33 = 1026.67 cm³. (For a scooped-out cavity you subtract; the leftover is exactly two-thirds of the cylinder, since a cone is one-third.)
Example 6 — Sphere from ratio: How many spheres of radius 1 cm equal the volume of one sphere of radius 3 cm?
Volume scales with r³, so the big sphere's volume is 3³ = 27 times a small one. Hence 27 spheres. This "volume scales as the cube of the linear ratio" idea is worth remembering for comparison questions.
Important Question Patterns
1. Single solid (2–3 marks): find CSA/TSA/volume from given dimensions; find a dimension given the area or volume.
2. Cone slant height (1–2 marks): l = √(r² + h²), then CSA/TSA.
3. Combination surface area (3 marks): add exposed surfaces only (remember hidden joining faces) — ice-cream cone, tent, capsule, toy.
4. Combination volume (3–4 marks): add the volumes of the parts; for a cavity (e.g. a cone scooped out of a cylinder) subtract.
5. Ratio/comparison (2–3 marks): volume of a cone vs cylinder vs sphere of the same radius; how many small solids fill a large one.
⚡ Quick Revision
- Cuboid V = lbh, TSA = 2(lb+bh+hl). Cube V = a³, TSA = 6a².
- Cylinder CSA = 2πrh, TSA = 2πr(r+h), V = πr²h.
- Cone CSA = πrl, TSA = πr(l+r), V = (1/3)πr²h; l = √(r²+h²).
- Sphere SA = 4πr², V = (4/3)πr³. Hemisphere CSA = 2πr², TSA = 3πr², V = (2/3)πr³.
- Cone volume = 1/3 of cylinder (same base & height).
- Combination surface area: add exposed faces only (joining faces hidden). Combination volume: add the parts (subtract for cavities).
- Two hemispheres of equal radius = one sphere.
- Removed from syllabus: frustum of a cone, conversion of solids.
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