## Common use::Cylinder (geometry)

### ::concepts

Cylinder::surface    Right::''r''    Geometry::section    ''h''::volume    Height::units    Circular::radius

Common use In common use a cylinder is taken to mean a finite section of a right circular cylinder, i.e., the cylinder with the generating lines perpendicular to the bases, with its ends closed to form two circular surfaces, as in the figure (right). If the ends are not closed, one obtains an open cylinder, whose surface is topologically equivalent to an open annulus.

If the cylinder has a radius r and length (height) h, then its volume is given by

V = πr2h

and its surface area is:

• the area of the top r2) +
• the area of the bottom r2) +
• the area of the side (rh).

Therefore, an open cylinder without the top or bottom has surface area (lateral area)

A = 2πrh.

The surface including the top and bottom as well as the lateral area is called a closed cylinder. Its surface area is

A = 2πr2 + 2πrh = 2πr(r + h) = πd(r + h),

where d is the diameter.

For a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a given surface area, the closed cylinder with the largest volume has h = 2r, i.e. the cylinder fits snugly in a cube (height = diameter).<ref>{{#invoke:citation/CS1|citation |CitationClass=citation }}.</ref>

### Volume

Having a right circular cylinder with a height h units and a base of radius r units with the coordinate axes chosen so that the origin is at the center of one base and the height is measured along the positive x-axis. A plane section at a distance of x units from the origin has an area of A(x) square units where

$A(x)=\pi r^2$

or

$A(y)=\pi r^2$

An element of volume, is a right cylinder of base area Awi square units and a thickness of Δix units. Thus if V cubic units is the volume of the right circular cylinder, by Riemann sums,

${Volume \; of \; cylinder}=\lim_{||\Delta \to 0 ||} \sum_{i=1}^n A(w_i) \Delta_i x$
$=\int_{0}^{h} A(y) \, dy$
$=\int_{0}^{h} \pi r^2 \, dy$
$=\pi\,r^2\,h\,$

Using cylindrical coordinates, the volume can be calculated by integration over

$=\int_{0}^{h} \int_{0}^{2\pi} \int_{0}^{r} s \,\, ds \, d\phi \, dz$
$=\pi\,r^2\,h\,$
Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section

### Surface area

The formula for finding the surface area of a cylinder is, with h as height, r as radius, and S as surface area is $S=2\pi rh+2\pi r^2$ Or, with B as base area and L as lateral area, $S=L+2B$

### Cylindric sections

Cylindric section.

Cylindric sections are the intersections of cylinders with planes. For a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a single straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two parallel line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, in a circle.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Eccentricity e of the cylindric section and semi-major axis a of the cylindric section depend on the radius of the cylinder r and the angle between the secant plane and cylinder axis α in the following way:

$e=\cos\alpha\,$
$a=\frac{r}{\sin\alpha}\,$

Cylinder (geometry) sections