3D Mensuration • Topic 2 of 4

Cylinder & Cone

A right circular cylinder (radius r, height h) has volume πr²h, curved surface area 2πrh, and total surface area 2πr(r + h) — the two flat circular ends add 2πr². A cone shares the base circle but tapers to a point: volume ⅓πr²h, slant height l = √(r²+h²), curved surface area πrl, and total surface area πr(r + l). The cone is exactly one-third of the cylinder on the same base and height — a fact CAT loves to test when a cone is carved out of or compared with a cylinder. Watch the difference between vertical height h and slant height l: CSA always uses l, never h. For hollow cylinders (pipes), surface area combines the outer and inner curved surfaces plus the two ring-shaped ends; volume of material is π(R² − r²)h. When a solid is open at one end (a cup, a tube), drop the corresponding circular face from the total area.

✅ Solved examples

1. A cylinder has radius 7 cm and height 10 cm. Find its volume (use π = 22/7).
V = πr²h = (22/7)×49×10 = 22×7×10 = 1540 cm³.
2. A cone has radius 6 cm and height 8 cm. Find its slant height and curved surface area (π).
l = √(6²+8²) = √100 = 10 cm. CSA = πrl = π×6×10 = 60π cm².
3. A solid cone of radius r and height h is exactly one-third of a cylinder of the same base and height. Verify with r = 3, h = 7 (π).
Cylinder = π×9×7 = 63π. Cone = ⅓×63π = 21π = ⅓π×9×7. Confirmed.
4. A cylindrical pipe has outer radius 5 cm, inner radius 4 cm and length 14 cm. Find the volume of metal (π = 22/7).
V = π(R²−r²)h = (22/7)(25−16)(14) = (22/7)(9)(14) = 22×9×2 = 396 cm³.

✏️ Practice — try these, take hints as needed

1. The CSA of a cylinder is 264 cm² and its height is 6 cm. Find the radius (π = 22/7).
CSA = 2πrh.
2×(22/7)×r×6 = 264.
Solve for r.
7 cm
2. A cone has CSA 65π cm² and radius 5 cm. Find its vertical height.
πrl = 65π ⇒ l = 13.
h = √(l²−r²).
√(169−25).
12 cm
3. The volumes of a cylinder and a cone with equal radius and height are in what ratio?
πr²h vs ⅓πr²h.
Cancel common factor.
1 vs ⅓.
3 : 1
4. A cylinder of radius 3 cm and height 12 cm is melted into a cone of radius 6 cm. Find the cone height.
Volume conserved.
π×9×12 = ⅓π×36×h.
108 = 12h.
9 cm
5. Find the total surface area of a closed cylinder of radius 7 cm and height 3 cm (π = 22/7).
TSA = 2πr(r+h).
2×(22/7)×7×(7+3).
44×10.
440 cm²

📝 Topic test — 8 questions

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