Ratios of cylinders · 3 marks

The radii of two cylinders are in the ratio 2:3, and their heights are in the ratio 3:2. Find

(a) the ratio of their volumes, and
(b) the ratio of their curved surface areas.

Answer: (a) Volumes 2 : 3 (b) Curved surface areas 1 : 1

Step-by-step solution

Idea: Let the radii be 2k and 3k and the heights 3m and 2m. Volume uses r²h; curved surface area uses rh.

(a) the ratio of their volumes, and

Let radii be 2k, 3k and heights 3m, 2m.½ mark
V₁V₂ = π(2k)²(3m)π(3k)²(2m) = 4 × 39 × 2 = 1218 = 23.1 mark
Ratio of volumes = 2 : 3.½ mark

2 : 3

(b) the ratio of their curved surface areas.

CSA₁CSA₂ = 2π(2k)(3m)2π(3k)(2m) = 66 = 1.½ mark
Ratio of curved surface areas = 1 : 1.½ mark

1 : 1

(a) 2 : 3 (b) 1 : 1

Check: Radii 2 and 3, heights 3 and 2: volumes 12π and 18π (2 : 3 ✓); CSAs 12π and 12π (1 : 1 ✓).

Answer to write in the exam

(a)

Radii 2k, 3k; heights 3m, 2m

V₁V₂ = π(2k)²(3m)π(3k)²(2m) = 1218 = 23

∴ Ratio of volumes = 2 : 3

(b)

CSA₁CSA₂ = 2π(2k)(3m)2π(3k)(2m) = 66

∴ Ratio of CSAs = 1 : 1

Common mistakes that cost marks

Forgetting to square the radius ratio for volume, which gives 1 : 1 instead of 2 : 3.
Squaring the height ratio as well: (23)² × (32)² = 1, giving 1 : 1. In πr²h only the radius is squared.

How this can come in the exam

MCQ (1 mark)

Two cylinders have equal heights and radii in the ratio 3 : 4. Their volumes are in the ratio

3 : 4
9 : 16
27 : 64
6 : 8

Show answer

(B) 9 : 16
Volume ∝ r² when h is equal: 9 : 16.

Try one yourself

The radii of two cylinders are in the ratio 1 : 2 and their heights in the ratio 4 : 1. Find the ratio of their volumes and of their curved surface areas.