3D Mensuration • Topic 3 of 4

Sphere & Hemisphere

A sphere of radius r has volume (4/3)πr³ and surface area 4πr² — there is no "curved versus total" split because a sphere is all one curved surface. A hemisphere is half a sphere: volume (2/3)πr³, curved surface area 2πr², and total surface area 3πr² (the 2πr² curved part plus the flat circular face πr²). The dominant CAT theme is recasting: a sphere melted into n smaller spheres conserves total volume, so R³ = n·r³ when the small spheres are equal. Because surface area scales as r² but volume as r³, breaking one big sphere into many small ones sharply increases total surface area even though volume is unchanged — the classic "how much more paint is needed" question. Also remember the largest sphere that fits inside a cube of side a has radius a/2, and a sphere inscribed in a cylinder of radius r needs height 2r.

✅ Solved examples

1. Find the volume and surface area of a sphere of radius 3 cm (π).
V = (4/3)π×27 = 36π cm³. SA = 4π×9 = 36π cm².
2. A solid hemisphere has radius 7 cm. Find its total surface area (π = 22/7).
TSA = 3πr² = 3×(22/7)×49 = 3×22×7 = 462 cm².
3. A sphere of radius 6 cm is melted and recast into spheres of radius 2 cm. How many small spheres are formed?
n = R³/r³ = 6³/2³ = 216/8 = 27 spheres.
4. The volumes of two spheres are in ratio 64 : 27. Find the ratio of their surface areas.
Volume ratio = (r₁/r₂)³ = 64/27 ⇒ r₁/r₂ = 4/3. Area ratio = (4/3)² = 16 : 9.

✏️ Practice — try these, take hints as needed

1. The surface area of a sphere is 616 cm². Find its radius (π = 22/7).
4πr² = 616.
r² = 616×7/(4×22) = 49.
r = √49.
7 cm
2. A hemispherical bowl has radius 5 cm. Find its volume (π).
V = (2/3)πr³.
(2/3)π×125.
250π/3.
(250/3)π cm³ ≈ 261.8 cm³
3. A sphere of radius 4 cm is melted into 8 equal spheres. Find the radius of each small sphere.
Volume conserved.
4³ = 8·r³.
r³ = 8.
2 cm
4. The largest sphere is carved from a cube of side 10 cm. Find the sphere volume (π).
Sphere radius = a/2.
r = 5.
(4/3)π×125.
(500/3)π cm³ ≈ 523.8 cm³
5. If the radius of a sphere is doubled, by what factor does its volume increase?
V ∝ r³.
2³.
New = 8× old.
8 times

📝 Topic test — 8 questions

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