Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Overview
Mensuration is formula-driven. Marks come from identifying the solid, picking the correct formula, and keeping units consistent. Use π = 22/7 unless told otherwise.
1. Standard Formulae
Cylinder: CSA = 2πrh TSA = 2πr(h + r) V = πr²h
Cone: CSA = πrl TSA = πr(l + r) V = ⅓πr²h (l = √(r² + h²))
Sphere: SA = 4πr² V = 4/3 πr³
Hemisphere: CSA = 2πr² TSA = 3πr² V = 2/3 πr³
Frustum: CSA = π(R + r)l V = ⅓πh(R² + r² + Rr)
2. Combined Solids
Split into known shapes → add surface areas of exposed faces (not hidden joins), and add or subtract volumes for solid or hollow parts.
Worked Examples
Example 1 — Cylinder volume
Radius 7 cm, height 10 cm.
V = πr²h = (22/7)(7²)(10) = 22 × 7 × 10 = 1540 cm³
Example 2 — Cone slant height and CSA
Radius 7 cm, height 24 cm.
l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm
CSA = πrl = (22/7)(7)(25) = 550 cm²
Example 3 — Sphere volume
Radius 3 cm.
V = 4/3 πr³ = 4/3 × (22/7) × 27 = (4 × 22 × 27)/(3 × 7) = 2376/7 ≈ 339.43 cm³
Example 4 — Hemisphere TSA
Radius 7 cm.
TSA = 3πr² = 3 × (22/7) × 49 = 3 × 22 × 7 = 462 cm²
Example 5 — Combined solid
A toy is a cone (r = 7, h = 24) on a hemisphere (r = 7). Find total volume.
Cone V = ⅓πr²h = ⅓ × (22/7) × 49 × 24 = 1232 cm³
Hemi V = 2/3 πr³ = 2/3 × (22/7) × 343 = 718.67 cm³
Total ≈ 1950.67 cm³
Common Mistakes
- Using TSA when only CSA is asked (or vice-versa).
- Forgetting to compute slant height l for a cone.
- Counting the hidden join faces of a combined solid.
- Mixing units (cm with m) in one problem.
Exam Tips & Revision Checklist
- Keep a one-page formula sheet; memorise the ⅓ for cone volume.
- Identify the solid before choosing the formula.
- Practised: cylinder, cone (slant + CSA), sphere, hemisphere, combined solid.
Frequently Asked Questions
Curved Surface Area (CSA) covers only the curved part; Total Surface Area (TSA) includes the curved part plus the flat base(s).