Sphere and hemisphere, solid and hollow
A sphere has surface area 4πr² and volume (4/3)πr³. A hemisphere is half a sphere: CSA 2πr² and TSA 3πr², since the flat circle adds πr².
Formulas for spheres and hemispheres
| Solid | Surface area | Volume |
|---|---|---|
| Sphere, radius r | 4πr² | (4/3)πr³ |
| Hollow sphere, outer R and inner r | Outer surface 4πR² | (4/3)π(R³ − r³) |
| Solid hemisphere, radius r | CSA 2πr²; TSA 3πr² | (2/3)πr³ |
| Hollow hemisphere, outer R and inner r | CSA 2π(R² + r²); TSA π(3R² + r²) | (2/3)π(R³ − r³) |
Where 3πr² and π(3R² + r²) come from
A hemisphere is half of a sphere, so its curved part is half of 4πr², that is 2πr². A solid hemisphere also has a flat circular face of area πr². So TSA = 2πr² + πr² = 3πr². A hollow hemisphere, like a bowl with some thickness, has an outer curved face 2πR², an inner curved face 2πr², and a flat ring at the rim of area π(R² − r²). Together: 2π(R² + r²) + π(R² − r²) = π(3R² + r²).
Example 1: a sphere
Find the surface area and volume of a sphere of radius 21 cm. (Take π = 22/7.)
- Given: r = 21 cm.
- Formula: surface area = 4πr².
- = 4 × 22/7 × 21 × 21 = 4 × 22 × 3 × 21
- = 5544 cm²
- Formula: volume = (4/3)πr³.
- = 4/3 × 22/7 × 21 × 21 × 21
- 21 ÷ 7 = 3 and 3 ÷ 3 = 1, so this is 4 × 22 × 21 × 21
- = 88 × 441 = 38808 cm³
Surface area = 5544 cm², volume = 38808 cm³
Example 2: a hollow brass bowl
A hollow hemispherical brass bowl has outer radius 10 cm and inner radius 9 cm. Find its curved surface area and total surface area. (Take π = 3.14.)
- Given: R = 10 cm, r = 9 cm.
- Formula: CSA = 2π(R² + r²).
- CSA = 2 × 3.14 × (100 + 81) = 6.28 × 181
- = 1136.68 cm²
- Formula: TSA = π(3R² + r²).
- TSA = 3.14 × (3 × 100 + 81) = 3.14 × 381
- = 1196.34 cm²
CSA = 1136.68 cm², TSA = 1196.34 cm²
Example 3: capacity of a hemispherical tank
A hemispherical water tank has inner radius 42 cm. How many litres of water can it hold? (Take π = 22/7.)
- Formula: volume of a hemisphere = (2/3)πr³.
- V = 2/3 × 22/7 × 42 × 42 × 42
- 42 ÷ 7 = 6 and 6 ÷ 3 = 2, so V = 2 × 22 × 2 × 42 × 42
- = 88 × 1764 = 155232 cm³
- Litres = 155232 ÷ 1000 = 155.232
The tank holds 155.232 litres (about 155 litres).
A laddu is nearly a sphere, and a steel bowl or a kadai is roughly a hemisphere. The volume of a hemispherical bowl tells how much payasam it can hold.
1 marks: “Write the total surface area of a solid hemisphere of radius r.” Curved part 2πr² + flat circle πr². Answer: 3πr².
2 marks: “The radii of two spheres are in the ratio 2 : 3. Find the ratio of their surface areas and of their volumes.” Surface areas go as r²: 2² : 3² = 4 : 9. Volumes go as r³: 2³ : 3³ = 8 : 27.
5 marks: “A hollow hemispherical bowl has outer radius 6 cm and inner radius 5 cm. Find its curved surface area, total surface area and the volume of material in it. (π = 3.14)” CSA = 2π(R² + r²) = 6.28 × 61 = 383.08 cm². TSA = π(3R² + r²) = 3.14 × 133 = 417.62 cm². Volume = (2/3)π(R³ − r³) = 2/3 × 3.14 × 91 ≈ 190.49 cm³.
Sphere: 4 and 4/3. Hemisphere halves them: 2 and 2/3. Add one πr² lid and the TSA becomes 3πr².