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Cylinder (geometry)

A cylinder is a three-dimensional shape generated by parallel lines around an axis; includes definitions, common formulas for volume and surface area, variants (circular, elliptic, parabolic), history and uses.

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Overview

A cylinder is a fundamental three-dimensional shape formed when all points at a fixed distance from a given line segment are collected. The line segment that runs through the center of the figure is the axis, and the continuous outer envelope is often called the surface. In everyday language the term covers both the hollow curved surface and the solid body enclosed by that surface; a circular cross-section gives the familiar right circular cylinder often seen in cans, pipes and rollers, which can be thought of as a special case of a circular prism.

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Definitions and basic properties

Standard parameters used to describe a cylinder are the radius r of its base and the height h measured along the axis. For a finite right circular cylinder the lateral (or curved) surface area equals 2πrh, while the total surface area adds the areas of the two circular bases to give 2πr(h + r). The enclosed volume is πr²h. These measures — surface area and surface formulas and the volume formula — are elementary and widely applied in engineering and mathematics.

Geometric varieties and characterization

From a more general viewpoint a cylinder can be defined as a ruled surface generated by a one-parameter family of parallel lines. In that setting a cylinder need not have circular bases: the cross section of the generating plane can be any plane curve. This broader notion appears in differential geometry where the defining straight rulings are required to be parallel, producing a surface that translates a given curve along a fixed direction.

Types and examples

Common named types are distinguished by the shape of the planar cross section. An elliptic cylinder uses an ellipse as its cross section and reduces to the circular case when the ellipse is a circle. A parabolic cylinder is generated by translating a parabola, and a hyperbolic cylinder by translating a hyperbola. These special forms are useful in analytic geometry and in solving partial differential equations, where symmetry or separability simplifies calculations.

History and mathematical significance

Formulas for the areas and volumes of cylindrical solids have been known since antiquity and were refined as part of the development of classical geometry. Ancient technicians and mathematicians used cylindrical shapes for containers and architectural elements; later work in calculus and solid geometry made the cylinder a standard example in integration, centroid and moment of inertia problems. The relation between a sphere and its circumscribed cylinder attracted particular attention in classical Greek mathematics.

Applications and distinctions

Cylinders are ubiquitous in engineering, manufacturing and everyday objects: pressure vessels, rollers, pistons, tanks and pipes rely on cylindrical geometry because of rotational symmetry and structural efficiency. In mathematics the distinction between a right cylinder (axis perpendicular to the base), an oblique cylinder (axis slanted relative to the base), and infinite cylinders (unbounded along the axis) is important for modeling and computation. Methods such as cylindrical coordinates and the cylindrical shell integration technique exploit the shape's symmetry to simplify problems in multivariable calculus and physics.

  • Key formulas: volume = πr²h; lateral area = 2πrh; total area = 2πr(h + r).
  • Variants: elliptic, parabolic, hyperbolic and oblique cylinders serve different analytic purposes.
  • Broader contexts: the ruled-surface definition connects classical solids to modern differential geometry and applied modeling.

Vertical circular cylinder: height h, radius r

Circular cylinder

In practice, the vertical circular cylinder plays an important role in various variations. Therefore concrete formulas are given for this.

Vertical circular cylinder

This results in

  • the volume V=\pi \;r^{2}\;h\ ,(base area × height)
  • the lateral surface M=2\pi r\;h\ ,(the unwinding is a rectangle of length 2\pi rand height h)
  • the surface O=2\pi r^{2}+2\pi rh\ .

A straight circular cylinder with {\displaystyle h=2r}is called an equilateral cylinder. This name is explained as follows: If you intersect such a cylinder with a plane containing the cylinder axis, you get a square (with side length 2r).

If the cross section is an ellipse with semiaxes a,b, then

  • V=\pi ab\;h\ .There is no simple formula for the sheath area.

Hollow cylinder

If a straight circular cylinder has a bore along its axis, it is called a hollow cylinder. For a hollow cylinder - such as a straight piece of pipe - the determining quantities are, in addition to the height , hthe outer radius Rand the inner radius r. The wall thickness b is therefore R-r.

  • The volume is V=\pi R^{2}h-\pi r^{2}h=\pi (R^{2}-r^{2})\;h\ ,
  • the lateral surface (inner and outer) M=2\pi (R+r)\;h\ ,
  • the surface O=2\pi (R^{2}-r^{2})+2\pi (R+r)\;h=2\pi (R+r)(R-r+h)\ .

If the height hof a hollow cylinder is smaller than its outer radius Rwe speak of a perforated disc with a concentric, circular opening.

Cylinder section

If you intersect a straight circular cylinder (radius r) with a plane at an angle, the intersection curve is an ellipse. If the lower cylinder section has the minimum height h_{1}and the maximum height h_{2}then the intersection ellipse has

  • the major semiaxis a={\sqrt {r^{2}+({\tfrac {h_{2}-h_{1}}{2}})^{2}}}and the minor semiaxis {\displaystyle b=r}, where {\displaystyle {\tfrac {h_{2}-h_{1}}{2}}=r\tan {\beta }}, with β \beta the inclination angle of the section plane,
  • the numerical eccentricity ε {\displaystyle \varepsilon =\sin \beta }.

The cylinder section itself has

  • the volume {\displaystyle V=\pi r^{2}({\tfrac {h_{1}+h_{2}}{2}})},
  • the lateral surface {\displaystyle M=2\pi r({\tfrac {h_{1}+h_{2}}{2}})=\pi r(h_{1}+h_{2}),}
  • the surface {\displaystyle O=\pi (r^{2}+ar)+\pi r(h_{1}+h_{2})=\pi r(r+a+h_{1}+h_{2})}.

Note: The volume and the surface area are equal to those of the cylinder with mean height {\tfrac {h_{1}+h_{2}}{2}}.

Volume calculation of a lying circular cylinder (tank problem)

The calculation of the content V of a partially filled lying circular cylinder can be done using the length L, the radius rand the filling height h. According to the equation volume = base area - height given above, the volume of the filling is obtained by multiplying the area of the circle segment by the length Lthe cylinder:

V=r^{2}L\left(\arccos \left({\frac {r-h}{r}}\right)-(r-h){\frac {{\sqrt {2rh-h^{2}}}}{r^{2}}}\right).

Straight circular cylinder with unwound jacket
Straight circular cylinder with unwound jacket
Hollow cylinder
Hollow cylinder
obliquely cut straight circle cylinder
obliquely cut straight circle cylinder
Partially filled horizontal cylinder (tank)
Partially filled horizontal cylinder (tank)

General cylinder

In mathematics, one defines a cylinder (mantle) more generally:

  • A plane curve c_{0}in a plane ε \varepsilon _{0}is {\vec {a}}shifted along a straight line not \varepsilon _{0}contained in ε by a fixed distance Each two corresponding points of the curves c_{0}and the shifted curve c_{1} are connected by a line. The set of these parallel lines forms the associated cylinder surface (see figure). The curve c_{0}is called a directrix. A straight line lying on the cylinder is called the generatrix or generatrix line.

If the curve is a circle, the result is a skew circular cylinder. If {\vec a}\perp \varepsilon _{0}, the result is a perpendicular circular cylinder.

If c_{0}a closed curve, the surface with the two boundary surfaces can again be understood as the surface of a body. If the curve c_{0}not closed, e.g. a parabolic arc (see below), the cylinder is just the lateral surface explained above, which however can be part of a surface of a body.

The geometrical peculiarity of a cylindrical surface consists in the following fact:

  • A cylindrical surface contains straight lines, it is a ruled surface, and can be unwound into the plane without distortion.

This property in particular makes the cylinder surface interesting for the manufacture of sheet metal cladding.

  • If the generating curve is a polygon, it is called a prism (see examples).
Definition of a general cylinder and example of an oblique circular cylinder
Definition of a general cylinder and example of an oblique circular cylinder
Examples of cylinders: circular cylinder and elliptical cylinder above, elliptical cylinder below: Prisms
Examples of cylinders: circular cylinder and elliptical cylinder above, elliptical cylinder below: Prisms

Questions and answers

Q: What is a cylinder?

A: A cylinder is a three dimensional geometric shape with the surface formed by points at a fixed distance from a given line segment, known as the axis of the cylinder. It can be thought of as a circular prism and both the surface and solid shape created inside can be called a cylinder.

Q: How long have people known about the surface area and volume of cylinders?

A: The surface area and volume of cylinders have been known since ancient times.

Q: What are elliptic, parabolic, and hyperbolic cylinders?

A: Elliptic, parabolic, and hyperbolic cylinders are cylinders whose cross section is an ellipse, parabola, or hyperbola respectively.

Q: How is a cylinder defined in differential geometry?

A: In differential geometry, a cylinder is defined more broadly as a ruled surface which is spanned by a one-parameter family of parallel lines.

Q: What does it mean for something to be "ruled"?

A: To be "ruled" means that it has straight lines drawn on it in some way or another.

Q: Is there only one type of cylinder?

A: No, there are many different types of cylinders such as elliptic, parabolic, and hyperbolic cylinders which all have different cross sections.

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