Cylinders

Peel the label off a tin can and it lies flat as a rectangle whose width is the can's circumference, 2πr.

Cut a paper tube straight down

Find an empty toilet-roll tube and a pair of scissors. Cut once, straight down from top to bottom, and flatten it. You get a rectangle. Its height is the height of the tube. Its width? That is the distance once around the circle, the circumference, 2πr. The model does the same when you press Unfold: the curved side opens at the back seam, keeps its width while the curve straightens out, and finishes with the two circles touching the long edges of the rectangle.

The side is a rectangle 2πr wide and h tall

So the curved surface, the part that stops most people, has area 2πr × h = 2πrh. Add the two circles at the ends and you have the whole surface: A = 2πr² + 2πrh, which can also be written 2πr(r + h). The model's cylinder starts at r = 2 cm and h = 5 cm, so its surface is 8π + 20π = 28π ≈ 87.96 cm².

Real objects do not always have both circles. A cup without a lid has only a bottom, so its surface is πr² + 2πrh. A pipe or a cardboard tube has no circles at all, and only 2πrh counts. In a real problem, half the work is reading which surfaces are actually there.

Back to labels. A label for a can of radius 4 cm and height 10 cm has area 2π × 4 × 10 = 80π ≈ 251.3 cm². It must be 2π × 4 ≈ 25.1 cm wide, and a little more in practice so the ends can overlap and be glued. The same rectangle turns up whenever something is wrapped around a cylinder: the card inside a toilet roll, a sheet of metal rolled into a pipe, a poster round a column.

Volume follows the prism rule with a circle for a base

A cylinder has two identical, parallel circular bases and the same cross-section all the way between them. That is the prism idea again, so V = base area × height = πr²h. For r = 2 cm and h = 5 cm, V = π × 4 × 5 = 20π ≈ 62.83 cm³. The symbol π is the ratio of any circle's circumference to its diameter, about 3.14159. Leave answers as multiples of π, like 20π, and turn them into decimals only at the end, so that rounding errors have no chance to pile up along the way.

Radius matters more than height. Double the height and the volume doubles. Double the radius and the base area becomes four times larger, so the volume does too. Set r = 2 in the model and note the volume, then try r = 4 with the same height. This is why a slightly wider cup holds much more than a slightly taller one.