Single solids: the formulas you need
Before joining solids, know each one alone: its curved surface area, total surface area and volume. A cone also needs its slant height l = √(r² + h²).
Three words to know
Curved surface area (CSA) is the area of the curved part only, like the label on a tin. Total surface area (TSA) is the area of all the outside surfaces, curved and flat. Volume is the space a solid fills, like the water a bottle holds. Area is in square units (cm², m²). Volume is in cubic units (cm³, m³). Remember 1000 cm³ = 1 litre and 1 m³ = 1000 litres.
Formula sheet for single solids
| Solid | Curved or lateral surface area | Total surface area | Volume |
|---|---|---|---|
| Cuboid (length l, breadth b, height h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
| Cube (edge a) | 4a² | 6a² | a³ |
| Cylinder (radius r, height h) | 2πrh | 2πr(r + h) | πr²h |
| Cone (radius r, height h, slant height l) | πrl | πr(l + r) | (1/3)πr²h |
| Sphere (radius r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (radius r) | 2πr² | 3πr² | (2/3)πr³ |
Worked example 1: a cone
A cone has base radius 6 cm and height 8 cm. Find its slant height, curved surface area and volume. (Use π = 3.14.)
- Given: r = 6 cm, h = 8 cm.
- Formula: l = √(r² + h²).
- l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
- Formula: CSA = πrl.
- CSA = 3.14 × 6 × 10 = 188.4 cm².
- Formula: volume = (1/3)πr²h.
- Volume = (1/3) × 3.14 × 36 × 8 = 3.14 × 96 = 301.44 cm³.
l = 10 cm, CSA = 188.4 cm², volume = 301.44 cm³.
Worked example 2: a steel katori
A steel bowl (katori) is a hemisphere of inner radius 7 cm. Find the area of its inner curved surface and how much dal it can hold. (Use π = 22/7.)
- Given: r = 7 cm.
- Formula: CSA of a hemisphere = 2πr².
- CSA = 2 × (22/7) × 7 × 7 = 2 × 22 × 7 = 308 cm².
- Formula: volume of a hemisphere = (2/3)πr³.
- Volume = (2/3) × (22/7) × 7 × 7 × 7 = (2 × 22 × 49) ÷ 3 = 2156 ÷ 3 ≈ 718.67 cm³.
- In litres: 718.67 ÷ 1000 ≈ 0.72 litre (rounded to 2 decimal places).
Inner curved surface = 308 cm²; the bowl holds about 718.67 cm³, nearly 0.72 litre.
Hemisphere: CSA or TSA?
In a cone, l means slant height. In a cuboid, l means length. Read the question to see which one is meant.
A kulfi mould is a cone, a ladoo is close to a sphere and a steel katori is close to a hemisphere. Tailors, painters and tank makers use these formulas to buy just enough cloth, paint or metal sheet.
1 marks: “Write the total surface area of a solid hemisphere of radius r.” 3πr²: the curved part 2πr² plus the flat circle πr².
2 marks: “A cone has radius 8 cm and height 15 cm. Find its curved surface area in terms of π.” l = √(64 + 225) = √289 = 17 cm. CSA = πrl = π × 8 × 17 = 136π cm².
3 marks: “Find the volume and the total surface area of a solid hemisphere of radius 21 cm. (Use π = 22/7.)” Volume = (2/3) × (22/7) × 21 × 21 × 21 = 19404 cm³. TSA = 3πr² = 3 × (22/7) × 441 = 4158 cm².
A hemisphere is half a sphere: half of 4πr² is 2πr² (curved), and one flat lid πr² makes 3πr² (total). Its volume is half of (4/3)πr³, that is (2/3)πr³.