Manufacturers want minimum metal for maximum soup. For a fixed volume, what shape uses the least material?
Two ends and a wrapped label
A can's metal is three pieces: the top disc (πr²), the bottom disc (πr²), and the curved side, which unrolls into a rectangle 2πr wide and h tall (2πrh). Total: 2πr² + 2πrh. Hold the volume fixed and shrink that total — calculus says the sweet spot is when the height equals the diameter, h = 2r.
Real cans are usually a bit taller than 2r — labels read better, and tall cans stack and grip nicer. Maths hands you the optimum; packaging engineers trade a little metal for everything else.
A can has radius 5 cm and height 12 cm. How much metal (surface area) does it use? (Use π ≈ 3.14.)
Height equals diameter. Calculus says this minimises surface area for a given volume.
Real cans are usually taller than 2r — because labels and stacking matter too. Maths gives the optimum; reality compromises.