Cylinders

A right circular cylinder has its axis at right angles to its circular base: a pipe, a biscuit tin. Any slice parallel to the base is a circle of the same size.

Curved surface: unroll it. Cut the curved surface straight down its side and lay it flat: it is a rectangle, 2πr long (the circumference) and h high. So CSA = 2πrh.

Add the ends as needed:

• closed at both ends: TSA = 2πrh + 2πr² = 2πr(h + r);
• open at one end, like a tumbler: πr(2h + r).

Volume = base area × height = πr²h. Picture a stack of thin round khakhras: each is a circle of area πr², and the stack is h tall.

Why r matters more. Doubling r and halving h keeps the curved surface the same, but doubles the volume, because r is squared in the volume.

Melting and recasting keeps the volume. A 12 cm cube (1728 cm³) makes 11 complete rods of 150.86 cm³ each.