# Cylinder (geometry)

Vertical circular cylinder: height , radius${\ displaystyle h}$${\ displaystyle r}$

A cylinder (Latin cylindrus, ancient Greek κύλινδρος kýlindros , from κυλίνδειν kylíndein , 'roll', 'wallow') is in the simplest case one

• Area whose points have the same distance from a fixed straight line, the axis .${\ displaystyle r}$

Since such an area is infinitely extended, it is usually cut with two parallel planes of distance (see picture). ${\ displaystyle h}$

• If the cutting planes are perpendicular to the axis, a vertical (or straight) circular cylinder with radius and height is created . The surface trimmed in this way is called the outer surface of the cylinder, the cut surfaces perpendicular to the axis can each be referred to as the base surface .${\ displaystyle r}$${\ displaystyle h}$

Since a straight circular cylinder can also be imagined to be generated by rotating a distance around the (parallel) cylinder axis, it is also called a rotary cylinder . The generating lines are called the surface lines of the cylinder or generating lines .

In engineering , a cylinder is often understood to be the body that is enclosed by the lateral surface and the two circular cutting surfaces.

In mathematics , a cylinder is defined more generally (see section on general cylinders )

## Circular cylinder

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

### Vertical circular cylinder

Straight circular cylinder with unwound jacket

It arises for

• the volume (base × height)${\ displaystyle V = \ pi \; r ^ {2} \; h \,}$
• the outer surface (the development is a rectangle of length and height )${\ displaystyle M = 2 \ pi r \; h \,}$${\ displaystyle 2 \ pi r}$${\ displaystyle h}$
• the surface ${\ displaystyle O = 2 \ pi r ^ {2} +2 \ pi rh \.}$

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

If the cross-section is an ellipse with the semiaxes , then is ${\ displaystyle a, b}$

• ${\ displaystyle V = \ pi from \; h \.}$ There is no simple formula for the surface area.

### Hollow cylinder

Hollow cylinder

If a straight circular cylinder has a bore along its axis, it is called a hollow cylinder . For a hollow cylinder - for example a straight piece of pipe - the determining parameters are the height, the outer radius and the inner radius . The wall thickness b is thus . ${\ displaystyle h}$${\ displaystyle R}$${\ displaystyle r}$ ${\ displaystyle Rr}$

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

If the height of a hollow cylinder is smaller than its outer radius , it is referred to as a perforated disc with a concentric , circular opening. ${\ displaystyle h}$${\ displaystyle R}$

### Cylinder section

obliquely cut straight circular cylinder

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

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

The cylinder section itself has

• the volume ,${\ displaystyle V = \ pi r ^ {2} ({\ tfrac {h_ {1} + h_ {2}} {2}})}$
• the outer 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 the same as that of the cylinder with the middle height . ${\ displaystyle {\ tfrac {h_ {1} + h_ {2}} {2}}}$

### Volume calculation of a horizontal circular cylinder (tank problem)

Partially filled horizontal cylinder (tank)

The calculation of the content of a partially filled horizontal circular cylinder can be made on the basis of the length , the radius and the filling level . According to the above equation volume = base · height , the volume of filling is calculated by multiplying the surface area of the circle segment with the length of the cylinder: ${\ displaystyle V}$${\ displaystyle L}$${\ displaystyle r}$${\ displaystyle h}$${\ displaystyle L}$

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

## General cylinder

Definition of a general cylinder and an example of an oblique circular cylinder
Examples of cylinders: above circular cylinder and elliptical cylinder, below: prisms

In mathematics, a cylinder (shell) is defined more generally:

• A plane curve in a plane is shifted a fixed distance along a straight line that is not contained in. Two corresponding points of the curves and the shifted curve are connected by a segment. The entirety of these parallel lines forms the associated cylinder surface (see picture). The curve is called the guide curve . A straight line lying on the cylinder is called the generating line or surface line .${\ displaystyle c_ {0}}$${\ displaystyle \ varepsilon _ {0}}$${\ displaystyle \ varepsilon _ {0}}$${\ displaystyle {\ vec {a}}}$${\ displaystyle c_ {0}}$${\ displaystyle c_ {1}}$${\ displaystyle c_ {0}}$

If the curve is a circle, an oblique circular cylinder is created . If is, the result is a vertical circular cylinder. ${\ displaystyle {\ vec {a}} \ perp \ varepsilon _ {0}}$

Is a closed curve, may be regarded as a back surface of a body, the peripheral surface with the two boundary surfaces. If the curve is not closed, e.g. B. a parabolic arc (see below), the cylinder is only the outer surface explained above, which can, however, be part of a surface of a body. ${\ displaystyle c_ {0}}$${\ displaystyle c_ {0}}$

The geometric peculiarity of a cylinder surface consists in the following fact:

• A cylindrical surface contains straight, it is a ruled surface , and can not distorted in the plane of handled be.

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

• If the generating curve is a polygon, one speaks of a prism (see examples).

## Properties of a general cylinder

Leaning cylinder: designations
oblique elliptical cylinder in general position
vertical circular cylinder in a general position
parabolic cylinder
hyperbolic cylinder

Volume, lateral surface and surface of a general cylinder are calculated as follows:

• Volume : if is a closed curve,${\ displaystyle V = G \ cdot h \,}$${\ displaystyle c_ {0}}$
where is the base area (of the enclosed area) and the height (see Cavalier principle ).${\ displaystyle G}$${\ displaystyle c_ {0}}$${\ displaystyle h}$

In the case of a prism, the base area can either be calculated directly (rectangle) or by a suitable division into triangles and / or rectangles (see area ). If the curve is piecewise smooth, the content can be determined directly or numerically using suitable integrals. ${\ displaystyle G}$${\ displaystyle c_ {0}}$

• Outer surface :${\ displaystyle M = U \ cdot l \,}$
where the circumference ( arc length ) of the cross-section (intersection curve to the surface lines) and the length of the surface (see picture). Note: can be understood as a vertical parallel projection of the guide curve on any cross-sectional plane (perpendicular to the surface lines).${\ displaystyle U}$${\ displaystyle c _ {\ perp}}$${\ displaystyle \ perp}$${\ displaystyle l}$${\ displaystyle c _ {\ perp}}$${\ displaystyle c_ {0}}$

For a vertical cylinder, and is the length of the guide curve . ${\ displaystyle l = h}$${\ displaystyle U}$${\ displaystyle c_ {0}}$

For an inclined cylinder, the height is where is the angle of the cylinder axis (direction from ) and the normal of the plane . In the case of an oblique circular or elliptical cylinder, the cross-sectional curve is an ellipse, in the case of a prism a polygon. The scope is simply in a polygon the sum of the edge lengths in a circle . In the case of a piecewise smooth guide curve , one can try to calculate the length of the cross-sectional curve with the help of a curve integral . But even with an ellipse that is not a circle, this is already a problem (see elliptic integral ) that can only be solved numerically. ${\ displaystyle h}$${\ displaystyle l = {\ tfrac {h} {\ cos \ varphi}} \,}$${\ displaystyle \ varphi}$${\ displaystyle {\ vec {a}}}$${\ displaystyle \ varepsilon _ {0}}$${\ displaystyle c _ {\ perp}}$ ${\ displaystyle U}$${\ displaystyle 2 \ pi r}$${\ displaystyle c_ {0}}$${\ displaystyle c _ {\ perp}}$

• Surface : if is a closed curve.${\ displaystyle O = M + 2 \ cdot G}$${\ displaystyle c_ {0}}$

## Analytical description

The outer surface of a vertical circular cylinder with radius and height , which is on the xy plane and has the z-axis as its axis, can be described by an equation in x, y and an inequality for z: ${\ displaystyle R}$${\ displaystyle h}$

• ${\ displaystyle x ^ {2} + y ^ {2} = R ^ {2}, \ 0 \ leq z \ leq h \,}$

If you want to the full cylinder describe, you have to by having to replace. ${\ displaystyle R}$${\ displaystyle r}$${\ displaystyle 0 \ leq r \ leq R}$

If the equation of a circle is replaced by the equation of an ellipse , the description of a vertical elliptical cylinder is obtained:

• ${\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} = 1, \ quad 0 \ leq z \ leq h \.}$ The volume is ${\ displaystyle V = \ pi from \; h \.}$

A parametric representation of a vertical circular or elliptical cylinder is obtained by using the usual parametric representation of a circle or an ellipse:

• ${\ displaystyle {\ vec {x}} (\ varphi, z) = (R \ cos \ varphi, R \ sin \ varphi, z), \ quad 0 \ leq \ varphi <2 \ pi, \ 0 \ leq z \ leq h}$
• ${\ displaystyle {\ vec {x}} (\ varphi, z) = (a \ cos \ varphi, b \ sin \ varphi, z), \ quad 0 \ leq \ varphi <2 \ pi, \ 0 \ leq z \ leq h \.}$

It is difficult to give the equation of a cylinder that is freely supported in space. The parametric representation of any elliptical cylinder, on the other hand, is relatively simple:

• ${\ displaystyle {\ vec {x}} (\ varphi, t) = {\ vec {q}} _ {0} + {\ vec {f}} _ {1} \ cos \ varphi + {\ vec {f }} _ {2} \ sin \ varphi + {\ vec {f}} _ {3} t, \ quad 0 \ leq \ varphi <2 \ pi, \ 0 \ leq t \ leq 1 \.}$

Here, the center of the bottom ellipse and three linearly independent vectors. points in the direction of the cylinder axis (see picture). ${\ displaystyle {\ vec {q}} _ {0}}$${\ displaystyle {\ vec {f}} _ {1}, {\ vec {f}} _ {2}, {\ vec {f}} _ {3}}$${\ displaystyle {\ vec {f}} _ {3}}$

If the three vectors are orthogonal in pairs and is , then the parametric representation describes a vertical circular cylinder with radius and height (see figure). ${\ displaystyle {\ vec {f}} _ {1}, {\ vec {f}} _ {2}, {\ vec {f}} _ {3}}$${\ displaystyle | {\ vec {f}} _ {1} | = | {\ vec {f}} _ {2} | = R}$${\ displaystyle R}$${\ displaystyle | {\ vec {f}} _ {3} |}$

The fact that any elliptical cylinder always contains circles is shown in the circular section plane.

This type of parametric representation is very flexible. E.g. provides

• ${\ displaystyle {\ vec {x}} (s, t) = {\ vec {q}} _ {0} + {\ vec {f}} _ {1} s + {\ vec {f}} _ {2 } s ^ {2} + {\ vec {f}} _ {3} t, \ quad -s_ {0} \ leq s \ leq s_ {0}, \ 0 \ leq t \ leq 1 \.}$

represents a parabolic cylinder in a general position (see picture, parabola ).

A vertical parabolic cylinder can also be passed through in the same way as a vertical circular cylinder

• ${\ displaystyle y = ax ^ {2}, \ 0 \ leq z \ leq h \,}$

describe.

The parametric representation

• ${\ displaystyle {\ vec {x}} (s, t) = {\ vec {q}} _ {0} \ pm {\ vec {f}} _ {1} \ cosh s + {\ vec {f}} _ {2} \ sinh s + {\ vec {f}} _ {3} t, \ quad -s_ {0} \ leq s \ leq s_ {0}, \ 0 \ leq t \ leq 1 \.}$

represents a hyperbolic cylinder in a general position (see hyperbola ).

A vertical hyperbolic cylinder can be passed through analogously to the vertical elliptical cylinder

• ${\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} - {\ frac {y ^ {2}} {b ^ {2}}} = 1, \ 0 \ leq z \ leq H\ ,}$

describe.

## Application examples

### silo

cylindrical grain silos

Grain silos often have the shape of a cylinder.

A cylindrical silo with the diameter of 12 meters and the height is 60 meters to 40 percent with wheat filled. So it is and . ${\ displaystyle r = 6 \ \ mathrm {m}}$${\ displaystyle h = 0 {,} 4 \ cdot 60 \ \ mathrm {m} = 24 \ \ mathrm {m}}$

This gives the volume and the surface :

• Volume :${\ displaystyle V = \ pi \ cdot r ^ {2} \ cdot h = \ pi \ cdot (6 \ \ mathrm {m}) ^ {2} \ cdot 24 \ \ mathrm {m} \ approx 2714 \ \ mathrm {m ^ {3}}}$
• Surface :${\ displaystyle O = 2 \ cdot \ pi \ cdot r \ cdot (r + h) = 2 \ cdot \ pi \ cdot 6 \ \ mathrm {m} \ cdot (6 \ \ mathrm {m} +24 \ \ mathrm {m}) \ approx 1131 \ \ mathrm {m ^ {2}}}$

The grain silo is filled with around 2714 cubic meters of wheat . The surface is about 1131 square meters .

### Drinking glass

A roughly cylindrical drinking glass

Some drinking glasses have approximately the shape of a cylinder.

A cylindrical drinking glass with a diameter of 74 millimeters and a filling height of 92 millimeters is half filled with orange juice . So it is and . ${\ displaystyle r = 37 \ \ mathrm {mm}}$${\ displaystyle h = {\ tfrac {1} {2}} \ cdot 92 \ \ mathrm {mm} = 46 \ \ mathrm {mm}}$

This gives the volume and the surface :

• Volume :${\ displaystyle V = \ pi \ cdot r ^ {2} \ cdot h = \ pi \ cdot (37 \ \ mathrm {mm}) ^ {2} \ cdot 46 \ \ mathrm {mm} \ approx 198 \ cdot 10 ^ {3} \ \ mathrm {mm ^ {3}} = 198 \ \ mathrm {cm ^ {3}} = 198 \ \ mathrm {ml}}$
• Surface :${\ displaystyle O = 2 \ cdot \ pi \ cdot r \ cdot (r + h) = 2 \ cdot \ pi \ cdot 37 \ \ mathrm {mm} \ cdot (37 \ \ mathrm {mm} +46 \ \ mathrm {mm}) \ approx 193 \ cdot 10 ^ {2} \ \ mathrm {mm ^ {2}} = 193 \ \ mathrm {cm ^ {2}}}$

The drinking glass is filled with about 198 milliliters of orange juice . The surface is about 193 square centimeters .