Cylinder

A cylinder is a closed solid that has two parallel (usually circular) bases connected by a curved surface.

Try this In the figure below, drag the orange dot to vary the dimensions of the cylinder.

Base and side

A cylinder is a geometric solid that is very common in everyday life, such as a soup can. If you take it apart you find it has two ends, called bases, that are usually circular. The bases are always congruent and parallel to each other. If you were to 'unroll' the cylinder you would find the the side is actually a rectangle when flattened out. (See Surface area of a cylinder).

Height

The height h is the perpendicular distance between the bases. It is important to use the perpendicular height (or 'altitude') when calculating the volume of an oblique cylinder.

Radius

The radius r of a cylinder is the radius of a base. If you are given the diameter instead, remember to halve it.

Axis

A line joining the center of each base.

Right and oblique cylinders

When the two bases are exactly over each other and the axis is a right angles to the base, this is a called a 'right cylinder'. If one base is displaced sideways, the axis is not at right angles to the bases and the result is called an oblique cylinder. The bases, although not directly over each other, are still parallel.
In the applet at the top of the page, check the "allow oblique" box and drag the orange dot sideways to see an oblique cylinder.

Right Cylinder	Oblique Cylinder

Volume and surface area

Volume		See Volume of a cylinder for more.
Surface area		See Surface area of a cylinder for more.

The shape of the bases

Usually the bases are circles, so a familiar soup can would be technically called a 'right circular cylinder'. This is the most common kind, and if someone just says 'cylinder' this is usually what they mean. The bases can however be almost any curved shape, but the most common alternative to a circle is an ellipse. The shape would then be called an 'elliptical cylinder'.

Relation to a prism

A prism is a solid with bases that are polygons and the sides are flat surfaces. (See Definition of a prism). Strictly speaking a cylinder is not a prism, however it is extremely similar. If you imagine a prism with regular polygons for bases, as you increase the number of sides, the solid gets to look just like a cylinder. So we can say that a cylinder is a prism with an infinite number of faces.
In the applet on the right, keep clicking on 'more' to see this effect. This is similar to the way a regular polygon turns into a circle when the number of sides gets infinitely large. See Area of a circle derivation.
A cylinder is also similar to a prism in that

1. It has the same cross section anywhere.
2. Its volume is calculated by multiplying the area of a base by the height.
3. It can be right or oblique.

Another way to produce a cylinder

Start with just the axis and then position another line that is the same length, called the generatrix, parallel to it and distance r from it.
If you move the line so that it remains parallel to, and the same distance from the axis, it will trace out a circular cylinder. In the applet on the right, press Run to see the cylinder thus created.
The path traced out by one end of the line is called the directrix, and is usually a circle, but can be any curved path.

A way to think about this method is to consider the surface created as the locus of the generatrix. Just as the locus of a point is a line, so the locus of a line is a surface.

Author's note

This way of defining a cylinder is in many textbooks and is presented for completeness. However, it seems inadequate because it does not generate the bases of the cylinder. It defines instead a tube without ends. Normally a cylinder is defined as a closed surface which would by necessity include the two bases. This is also in conflict with the surface area of a cylinder which would usually include the area of the two bases.