Cylinder

The bare term cylinder often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an open cylinder. The formulae for the surface area and the volume of a right circular cylinder have been known from early antiquity.

A right circular cylinder can also be thought of as the solid of revolution generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.[3]

A tall and thin needle cylinder has a height much greater than its diameter, whereas a short and wide disk cylinder has a diameter much greater than its height.

Cylindric section

A cylindric section is the intersection of a cylinder's surface with a plane. They are, in general, curves and are special types of plane sections. The cylindric section by a plane that contains two elements of a cylinder is a parallelogram.[4] Such a cylindric section of a right cylinder is a rectangle.[4]

A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section.[5] If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.

Cylindric sections of a right circular cylinder

For a right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element. The right sections are circles and all other planes intersect the cylindrical surface in an ellipse.[6] If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle.

In the case of a right circular cylinder with a cylindric section that is an ellipse, the 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 α between the secant plane and cylinder axis, in the following way:

${\displaystyle e=\cos \alpha ,}$

${\displaystyle a={\frac {r}{\sin \alpha }}.}$

If the base of a circular cylinder has a radius r and the cylinder has height h, then its volume is given by

V = πr²h.

This formula holds whether or not the cylinder is a right cylinder.[7]

This formula may be established by using Cavalieri's principle.

A solid elliptic cylinder with the semi-axes a and b for the base ellipse and height h

In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base having semi-major axis a, semi-minor axis b and height h has a volume V = Ah, where A is the area of the base ellipse (= πab). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive x-axis and A(x) = A the area of each elliptic cross-section, thus:

${\displaystyle V=\int _{0}^{h}A(x)dx=\int _{0}^{h}\pi abdx=\pi ab\int _{0}^{h}dx=\pi abh.}$

Using cylindrical coordinates, the volume of a right circular cylinder can be calculated by integration over

${\displaystyle =\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{r}s\,\,ds\,d\phi \,dz}$

${\displaystyle =\pi \,r^{2}\,h.}$

Having radius r and altitude (height) h, the surface area of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:

• the area of the top base: πr²
• the area of the bottom base: πr²
• the area of the side: 2πrh

The area of the top and bottom bases is the same, and is called the base area, B. The area of the side is known as the lateral area, L.

An open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area)

L = 2πrh.

The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore,

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

where d = 2r is the diameter of the circular top or bottom.

For a given volume, the right circular cylinder with the smallest surface area has h = 2r. Equivalently, for a given surface area, the right circular cylinder with the largest volume has h = 2r, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).[8]

The lateral area, L, of a circular cylinder, which need not be a right cylinder, is more generally given by:

L = e × p,

where e is the length of an element and p is the perimeter of a right section of the cylinder.[9] This produces the previous formula for lateral area when the cylinder is a right circular cylinder.

Hollow cylinder

A right circular hollow cylinder (or cylindrical shell) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis, as in the diagram.

Let the height be h, internal radius r, and external radius R. The volume is given by

${\displaystyle V=\pi (R^{2}-r^{2})h=2\pi \left({\frac {R+r}{2}}\right)h(R-r).}$.

Thus, the volume of a cylindrical shell equals 2π(average radius)(altitude)(thickness).[10]

The surface area, including the top and bottom, is given by

${\displaystyle A=2\pi (R+r)h+2\pi (R^{2}-r^{2}).}$.

Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.[11]

A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases

In the treatise by this name, written c. 225 BCE, Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a sphere by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and diameter. The sphere has a volume two-thirds that of the circumscribed cylinder and a surface area two-thirds that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius r is 4/3πr³ = 2/3 (2πr³). The surface area of this sphere is 4πr² = 2/3 (6πr²). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.

If AB > 0 this is the equation of an elliptic cylinder.[15] Further simplification can be obtained by translation of axes and scalar multiplication. If ${\displaystyle \rho }$ has the same sign as the coefficients A and B, then the equation of an elliptic cylinder may be rewritten in Cartesian coordinates as:

${\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1.}$

This equation of an elliptic cylinder is a generalization of the equation of the ordinary, circular cylinder (a = b). Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid.

If ${\displaystyle \rho }$ has a different sign than the coefficients, we obtain the imaginary elliptic cylinders:

${\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=-1,}$

which have no real points on them. (${\displaystyle \rho =0}$ gives a single real point.)

If A and B have different signs and ${\displaystyle \rho \neq 0}$, we obtain the hyperbolic cylinders, whose equations may be rewritten as:

${\displaystyle \left({\frac {x}{a}}\right)^{2}-\left({\frac {y}{b}}\right)^{2}=1.}$

Finally, if AB = 0 assume, without loss of generality, that B = 0 and A = 1 to obtain the parabolic cylinders with equations that can be written as:[16]

${\displaystyle {x}^{2}+2a{y}=0.}$

In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.