Surface area and volume: what they mean
Surface area = total area of all the outside faces. Unit: cm² or m². Volume = space inside. Unit: cm³ or m³. 1 litre = 1000 cm³, and 1 m³ = 1000 litres.
Curved (or lateral) surface area = only the side, without the top and bottom. Total surface area = everything outside.
Cuboid and cube
A cuboid has length l, breadth b, height h. It has 6 rectangle faces in 3 matching pairs.
- Total surface area = 2(lb + bh + hl)
- Lateral surface area (4 walls) = 2h(l + b)
- Volume = l × b × h
A cube is a cuboid with all sides a: total surface = 6a², lateral = 4a², volume = a³.
Cylinder
A cylinder has two circle ends (radius r) and height h. Unroll its side: it is a rectangle 2πr by h.
- Curved surface area = 2πrh
- Total surface area = 2πrh + 2πr² = 2πr(r + h)
- Volume = base area × height = πr²h
Cone
A right circular cone has a circle base (radius r), height h (tip straight down to the centre) and slant height l (tip to the edge of the base). By Pythagoras, l = √(r² + h²).
- Curved surface area = πrl
- Total surface area = πrl + πr² = πr(l + r)
- Volume = ⅓πr²h (one third of the matching cylinder)
Pyramid
A pyramid has a flat polygon base and triangle faces that meet at a tip. For a square pyramid with base side a and height h:
- Slant height of each triangle face = √(h² + (a/2)²)
- Lateral surface area = 4 × ½ × a × slant height
- Total surface area = a² + lateral surface area
- Volume = ⅓ × base area × height = ⅓a²h
This ⅓ rule works for every pyramid and cone: volume = ⅓ × base area × height.
Sphere and hemisphere
A sphere is a perfect ball of radius r. A hemisphere is half a ball.
- Sphere: surface area = 4πr², volume = ⁴⁄₃πr³
- Hemisphere: curved surface = 2πr², total surface = 3πr² (curved + flat circle), volume = ⅔πr³
A hollow bowl open at the top uses only the curved part, 2πr².
Board exam pattern
Mensuration carries 14 marks in CBSE Class 9 (2026-27). Common questions: find the canvas for a tent (cone), the cost of painting walls (cuboid lateral surface), the capacity of a tank in litres, and the tin for a bowl (hemisphere). Always convert units first and write cm² or cm³.
Try it at home
Roll a paper cone and a paper cylinder with the same circle base and height. Fill the cone with rice and pour it into the cylinder. Count the pours: you will need about 3. Then cut open an empty juice box and flatten it to see its six faces; measure them and add to get its surface area.
Key formulas and definitions
- Cuboid: TSA = 2(lb + bh + hl), LSA = 2h(l + b), V = lbh
- Cube: TSA = 6a², LSA = 4a², V = a³
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h
- Cone: l = √(r² + h²), CSA = πrl, TSA = πr(l + r), V = ⅓πr²h
- Pyramid: V = ⅓ × base area × h; square pyramid TSA = a² + 2a × slant height
- Sphere: SA = 4πr², V = ⁴⁄₃πr³
- Hemisphere: CSA = 2πr², TSA = 3πr², V = ⅔πr³
Worked examples
1. A box is 5 cm long, 4 cm wide and 3 cm high. Find its total surface area and volume.
TSA = 2(5×4 + 4×3 + 3×5) = 2(20 + 12 + 15) = 94 cm². Volume = 5 × 4 × 3 = 60 cm³.
2. A cube has side 6 cm. Find its lateral surface area, total surface area and volume.
LSA = 4 × 36 = 144 cm². TSA = 6 × 36 = 216 cm². Volume = 6³ = 216 cm³.
3. A cylinder has radius 7 cm and height 10 cm. Find its curved surface area, total surface area and volume. (π = 22/7)
CSA = 2 × 22/7 × 7 × 10 = 440 cm². Two ends = 2 × 22/7 × 49 = 308 cm². TSA = 748 cm². Volume = 22/7 × 49 × 10 = 1540 cm³.
4. A cone has radius 7 cm and height 24 cm. Find its slant height, curved surface area, total surface area and volume. (π = 22/7)
l = √(49 + 576) = √625 = 25 cm. CSA = 22/7 × 7 × 25 = 550 cm². TSA = 550 + 154 = 704 cm². Volume = ⅓ × 22/7 × 49 × 24 = 1232 cm³.
5. A square pyramid has base side 10 m and height 12 m. Find its total surface area and volume.
Slant height = √(12² + 5²) = 13 m. Each triangle = ½ × 10 × 13 = 65 m², four faces = 260 m². TSA = 100 + 260 = 360 m². Volume = ⅓ × 100 × 12 = 400 m³.
6. Find the surface area and volume of a ball of radius 7 cm, and the total surface area of a solid hemisphere of the same radius. (π = 22/7)
Sphere: 4 × 22/7 × 49 = 616 cm²; volume = ⁴⁄₃ × 22/7 × 343 ≈ 1437.33 cm³. Hemisphere TSA = 3πr² = 3 × 154 = 462 cm².
7. A cylindrical water tank has radius 1.4 m and height 2 m. How many litres does it hold? (π = 22/7)
Volume = 22/7 × 1.96 × 2 = 12.32 m³. 1 m³ = 1000 L, so it holds 12320 litres.
Common mistakes
- Using the height h instead of the slant height l in the cone's curved surface area πrl.
- Mixing units, such as radius in m and height in cm, or giving volume in cm² instead of cm³.
- Forgetting the flat circle in a solid hemisphere's total surface area (3πr², not 2πr²).
- Forgetting the ⅓ in the volume of a cone or pyramid.