Q.7 A rectangular paper sheet of dimensions 54 cm × 4 cm is taken. The two longer
edges of the sheet are joined together to create a cylindrical tube. A cube whose
surface area is equal to the area of the sheet is also taken.
Then, the ratio of the volume of the cylindrical tube to the volume of the cube is
(A) 1/π
(B) 2/π
(C) 3/π
(D) 4/π
cylinder and a cube from the same rectangular sheet
and comparing their volumes. The final ratio simplifies neatly to a constant
involving π.
Problem Overview
A rectangular paper sheet of dimensions 54 cm × 4 cm is:
- Rolled to form a cylinder by joining its longer edges
- Used to form a cube whose total surface area equals the area of the sheet
Find the ratio of the volume of the cylinder to the
volume of the cube.
Cylinder Dimensions
Joining the 54 cm edges forms the height of the cylinder:
h = 54 cm
The width 4 cm becomes the circumference of the base:
2πr = 4 ⟹ r = 2/π cm
Cylinder Volume
Volume of a cylinder:
Vcyl = πr²h
Substituting values:
Vcyl = π × (2/π)² × 54
= π × (4/π²) × 54
= 216/π cm³
Cube Dimensions
Area of the sheet:
54 × 4 = 216 cm²
This equals the total surface area of the cube:
6a² = 216
a² = 36 ⟹ a = 6 cm
Cube Volume
Vcube = a³ = 6³ = 216 cm³
Ratio Calculation
Volume ratio:
Vcyl / Vcube
= (216/π) / 216
= 1/π
Correct Answer
(A) 1/π
Option Analysis
(A) 1/π
✅ Correct. Matches exact volume calculation.
(B) 2/π
❌ Occurs if height is incorrectly taken as 4 cm or radius is miscalculated.
The longer edge (54 cm) must be the height.
(C) 3/π
❌ May result from confusing total surface area with lateral surface area.
(D) 4/π
❌ Arises if circumference or radius is doubled incorrectly.
Violates the given edge-joining condition.
Final Insight
Always identify which edge becomes the height and which becomes
the circumference when a sheet is rolled. Small interpretation
errors lead to large numerical mistakes in GATE aptitude questions.
