Cylinder and hollow cylinder
A cylinder has two circular ends and a curved side. Its curved surface area is 2πrh and volume is πr²h. A hollow cylinder uses radii R and r.
What a cylinder is
A right circular cylinder is the solid you get when a rectangle turns once around one of its sides. Its ends are two equal circles of radius r. Its height h is the distance between the two ends. Cut the curved side and open it out: you get a rectangle of length 2πr (the circumference of the base) and breadth h. So the curved surface area is 2πr × h.
Formulas for solid and hollow cylinders
| Measure | Solid cylinder | Hollow cylinder (outer radius R, inner radius r) |
|---|---|---|
| Curved surface area | 2πrh | 2π(R + r)h (outer and inner curved faces together) |
| Total surface area | 2πr(h + r) | 2π(R + r)(R − r + h) (curved faces plus the two rings) |
| Volume | πr²h | πh(R² − r²) (volume of the material) |
| Thickness | Not used | R − r |
Example 1: a closed tin
A closed cylindrical tin has radius 7 cm and height 20 cm. Find its curved surface area and total surface area. (Take π = 22/7.)
- Given: r = 7 cm, h = 20 cm.
- Formula: CSA = 2πrh.
- CSA = 2 × 22/7 × 7 × 20
- = 2 × 22 × 20 = 880 cm²
- Formula: TSA = 2πr(h + r).
- TSA = 2 × 22/7 × 7 × (20 + 7)
- = 44 × 27 = 1188 cm²
CSA = 880 cm², TSA = 1188 cm²
Example 2: a hollow iron roller
A hollow iron roller is 14 cm long. Its outer radius is 5 cm and its inner radius is 2 cm. Find its curved surface area, total surface area and the volume of iron in it. (Take π = 22/7.)
- Given: h = 14 cm, R = 5 cm, r = 2 cm.
- Formula: CSA = 2π(R + r)h.
- CSA = 2 × 22/7 × (5 + 2) × 14
- = 2 × 22/7 × 7 × 14 = 44 × 14 = 616 cm²
- Formula: TSA = 2π(R + r)(R − r + h).
- TSA = 2 × 22/7 × 7 × (5 − 2 + 14) = 44 × 17 = 748 cm²
- Formula: volume = πh(R² − r²).
- Volume = 22/7 × 14 × (25 − 4) = 44 × 21 = 924 cm³
CSA = 616 cm², TSA = 748 cm², volume of iron = 924 cm³
Mistakes to avoid
- If the question gives the diameter, halve it to get the radius first.
- An open cylinder, like a tumbler with no lid, has area 2πrh + πr²: one circle only.
- Change units before you calculate: 1 m = 100 cm, and 1 litre = 1000 cm³.
- Write cm² for area and cm³ for volume.
Rooftop water tanks, steel tumblers, gas cylinders and drainage pipes are cylinders. The volume tells how many litres a tank holds; the CSA tells how much paint or label its side needs.
1 marks: “The radius of a cylinder is doubled and its height is unchanged. How many times does its volume become?” V = πr²h. With radius 2r, V = π(2r)²h = 4πr²h. Answer: 4 times.
2 marks: “The curved surface area of a cylinder is 440 cm² and its radius is 7 cm. Find its height. (π = 22/7)” 2πrh = 440, so 2 × 22/7 × 7 × h = 440, that is 44h = 440. Answer: h = 10 cm.
5 marks: “A hollow cylindrical iron pipe is 21 cm long. Its outer and inner diameters are 8 cm and 6 cm. Find the volume of iron in it and its total surface area. (π = 22/7)” R = 4 cm, r = 3 cm, h = 21 cm. Volume = πh(R² − r²) = 22/7 × 21 × 7 = 462 cm³. TSA = 2π(R + r)(R − r + h) = 44 × 22 = 968 cm².
Open out the side of a cylinder and it is a rectangle, 2πr long and h high. So CSA = 2πrh, and volume = base area πr² × height.