Archimedes wanted a sphere inside a cylinder carved on his tomb
Around 225 BCE Archimedes proved that a sphere holds exactly two-thirds the volume of the snugly fitting cylinder around it, and has two-thirds of its surface area too. He was so proud of the discovery that he asked for a sculpted sphere and cylinder to mark his grave.
The word cylinder comes from the Greek kúlindros, meaning roller or tumbler. In school geometry it is a prism with a circular base, but mathematicians use the term more loosely. A cylindrical surface is traced by a straight line, the generatrix, sliding parallel to itself while always touching a fixed curve called the directrix. Cap that surface with two parallel planes and you get a solid cylinder, whose two bases are always congruent. If its sides stand perpendicular to the bases it is a right cylinder; otherwise it leans and is called oblique. Spin a rectangle around one of its edges and you produce a right circular cylinder, the everyday tin-can shape.
Slicing reveals variety. A plane cutting straight across a right circular cylinder gives a circle, one tilted at an angle gives an ellipse, and one running along the length through two of its lines gives a rectangle. When the cross-section at right angles is a parabola, ellipse or hyperbola instead of a circle, the cylinder takes that name.
Volume follows a simple rule: base area times height, and it works for leaning cylinders as well as upright ones, a fact that follows from Cavalieri's principle. An elliptic cylinder, for example, holds pi times its two semi-axes times its height. The surface-area and volume formulas for the right circular cylinder have been known since early antiquity. Engineers and packagers may like one neat result: for a fixed volume, the cylinder with the least surface area has a height equal to twice its radius, meaning it fits exactly inside a cube.
Hollow cylinders, or cylindrical shells, have a tidy volume too: two times pi, times the average radius, times the height, times the wall thickness. Both solid discs and shells serve as standard tools in calculus for finding volumes of solids of revolution.
Source: Cylinder
