A rectangular sheet of paper 88 cm × 10 cm is folded without overlapping to…

Concept

• When a rectangle is rolled into a cylinder without overlap, one side of the rectangle wraps around to form the circumference of the circular base and the perpendicular side becomes the height of the cylinder.

• For the circular base, circumference = 2 × π × r, so r = circumference ÷ (2 × π).

• For the cylinder, volume V = π × r2 × h, and capacity in litres = volume in cm3 ÷ 1000.

Application

The cylinder is stated to be 10 cm high, so the 10 cm side of the sheet is the height and the whole 88 cm side wraps around to form the base circumference. (Rolling the sheet the other way would make the height 88 cm, which contradicts the stated height.)

1. Base circumference = 88 cm and height h = 10 cm.

2. Circumference = 2 × π × r gives 2 × (22/7) × r = 88, i.e. (44/7) × r = 88.

3. r = 88 × 7 ÷ 44 = 14 cm.

4. V = π × r2 × h = (22/7) × 142 × 10.

5. (22/7) × 196 = 616, so V = 616 × 10 = 6160 cm3.

6. Capacity = 6160 ÷ 1000 = 6.16 litres, since 1000 cm3 = 1 litre.

Cross-check

The sheet becomes the curved surface of the cylinder, so its area must equal 2 × π × r × h: 88 × 10 = 880 cm2, and 2 × (22/7) × 14 × 10 = 880 cm2. This confirms r = 14 cm. A second route to the volume uses V = (curved surface area × r) ÷ 2 = (880 × 14) ÷ 2 = 6160 cm3, the same value.

Hence the capacity of the cylinder is 6.16 litres.