Volume Applications • Topic 3 of 3

Combined Solids

A combined solid is two or more standard shapes fused along a common face — a cone mounted on a cylinder (a pencil, a rocket), a hemisphere capping a cone (an ice-cream cone, a toy top), or a hemispherical dome on a cylindrical tower. Volume is simply additive: total V is the sum of the individual volumes, because melting nothing changes. Surface area is the subtle part. You add the curved surface areas (CSA) of the exposed parts, but the flat circular faces where two solids join are hidden inside and must NOT be counted. So a hemisphere on a cone has surface area πrl + 2πr², not πrl + 2πr² + 2(πr²) — the shared base circle disappears for both shapes. For a cone-on-cylinder toy that sits on a table, the visible surface is the cone’s CSA, the cylinder’s CSA, and the cylinder’s bottom circle, but never the join. The CAT-smart habit: sketch it, mark which circular faces are internal, and add only what light can touch. Often r is shared across the parts, so πr factors out neatly.

✅ Solved examples

1. A toy is a cone of radius 7 cm and height 24 cm mounted on a hemisphere of the same radius. Find its total volume.
Cone V = (1/3)π(7)²(24) = (1/3)π(49)(24) = 392π. Hemisphere V = (2/3)π(7)³ = (2/3)π(343) = 228.67π. Total = 392π + 228.67π = 620.67π ≈ 1950 cm³ (using π ≈ 22/7, ≈ 1950.67 cm³).
2. A solid is a cone (radius 3 cm) on top of a hemisphere (radius 3 cm), the cone’s slant height being 5 cm. Find the total surface area.
Cone CSA = πrl = π(3)(5) = 15π. Hemisphere CSA = 2πr² = 2π(9) = 18π. Shared circle is internal, so TSA = 15π + 18π = 33π ≈ 103.7 cm².
3. A cylindrical tank of radius 7 m and height 10 m is topped by a hemispherical dome of the same radius. Find the total volume.
Cylinder V = π(7)²(10) = 490π. Hemisphere V = (2/3)π(7)³ = (2/3)π(343) = 228.67π. Total = 718.67π ≈ 2258 m³ (π ≈ 22/7).
4. An ice-cream cone (radius 3.5 cm, height 12 cm) is filled and topped with a hemispherical scoop of the same radius. Find the volume of ice cream.
Cone V = (1/3)π(3.5)²(12) = (1/3)π(12.25)(12) = 49π. Hemisphere V = (2/3)π(3.5)³ = (2/3)π(42.875) = 28.583π. Total = 77.583π ≈ 243.8 cm³ (π ≈ 22/7 ⇒ ≈ 243.83 cm³).

✏️ Practice — try these, take hints as needed

1. A cylinder (radius 5 cm, height 12 cm) is topped by a hemisphere of radius 5 cm. Find the total volume in terms of π.
Add the two volumes.
πr²h + (2/3)πr³.
π(25)(12) + (2/3)π(125).
(300 + 250/3)π = 1150π/3 cm³ ≈ 1204 cm³
2. A hemisphere of radius 6 cm caps a cone of radius 6 cm and slant height 10 cm. Find the total surface area in terms of π.
Cone CSA = πrl; hemisphere CSA = 2πr².
Shared base circle is internal.
π(6)(10) + 2π(36).
132π cm² ≈ 414.7 cm²
3. A solid cylinder (radius 7 cm, height 20 cm) has a cone (same radius, height 9 cm) on top. Find total volume (π ≈ 22/7).
πr²h + (1/3)πr²H.
π(49)(20) + (1/3)π(49)(9).
980π + 147π.
1127π ≈ 3542 cm³
4. A wooden article is a cylinder (radius 3.5 cm, height 10 cm) with a hemisphere scooped OUT of each end. Find the volume removed (π ≈ 22/7).
Two hemispheres = one full sphere.
V = (4/3)πr³ with r = 3.5.
(4/3)π(42.875).
(4/3)(42.875)π ≈ 179.67 cm³
5. A capsule is a cylinder (radius 2 mm, length 8 mm between the flat faces) with a hemisphere on each end. Find the total volume in terms of π.
Cylinder + two hemispheres (= a sphere).
πr²h + (4/3)πr³.
π(4)(8) + (4/3)π(8).
(32 + 32/3)π = 128π/3 mm³ ≈ 134 mm³

📝 Topic test — 8 questions

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