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Cylinder

A cylinder is a three-dimensional solid bounded by two parallel congruent bases, which are planar closed curves, and a lateral surface generated by moving a straight line segment parallel to a fixed direction along the boundary of one base.^[1]^[2] In its most common form, known as a right circular cylinder, the bases are circles of equal radius, and the generating lines are perpendicular to the planes of the bases, resulting in circular cross-sections perpendicular to the axis.^[1]^[3] Cylinders can be classified into various types based on the shape of the bases and the orientation of the generating lines. A right cylinder has generating lines perpendicular to the bases, while an oblique cylinder features generating lines at an angle to the bases, leading to elliptical cross-sections.^[1] The bases may be circles (circular cylinder), ellipses (elliptic cylinder), parabolas (parabolic cylinder), or other curves, with the circular variant being the standard in elementary geometry.^[1]^[2] For a right circular cylinder with radius r and height h, the volume is given by V = \pi r^2 h, and the total surface area is $2\pi r (r + h), comprising the lateral area $2\pi r h and the two base areas $2\pi r^2.^[1] The mathematical study of cylinders dates back to ancient Greece, where Euclid defined them in his Elements as right circular solids with circular bases and perpendicular axes.^[3] Archimedes made significant contributions around 225 BCE in his treatise On the Sphere and Cylinder, proving that a sphere inscribed in a cylinder has two-thirds the volume and surface area of the circumscribing cylinder, a result he considered his greatest achievement and requested be depicted on his tombstone.^[4] Cylinders also appear in advanced mathematics, such as quadric surfaces in multivariable calculus, where their equations take forms like \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 for elliptic cylinders extending along the z-axis.^[5] Beyond pure geometry, cylinders model real-world objects like pipes, cans, and engine components, underscoring their practical importance.^[1]

Definition and Classification

Definition

In geometry, a cylinder is defined as a ruled surface generated by the motion of a straight line, called the ruling or generatrix, that moves parallel to a fixed direction while intersecting a fixed curve, known as the directrix, which lies in a plane not parallel to that direction.^[6] This construction ensures the surface consists entirely of straight lines, each parallel to the others, passing through corresponding points on the directrix.^[7] As a ruled surface, a cylinder possesses the property that every point on it lies on at least one straight line segment contained within the surface, with these rulings forming a family of parallel lines.^[8] The resulting surface extends infinitely in the direction of the rulings unless explicitly bounded by additional planes, distinguishing it from finite solids often visualized in applications.^[9] Unlike a cone, where the rulings converge to a single fixed point (the vertex), the rulings of a cylinder remain parallel and do not intersect.^[10] In contrast to a prism, which features polygonal bases connected by flat, parallelogram-shaped lateral faces, a general cylinder allows for a curved directrix, producing a non-polyhedral surface if the directrix is non-linear, though prisms can be viewed as special cases with straight-edged polygonal directrices and parallel bounding planes.^[11] A basic visualization of this concept arises when the directrix is a circle in a plane perpendicular to the fixed direction, yielding a right circular cylinder, where the rulings are perpendicular to the plane of the circle; more generally, cylinders may be right (rulings perpendicular to the directrix plane) or oblique (rulings at an angle).^[12]

Right versus oblique cylinders

A right cylinder is formed when the rulings, or generating lines, are perpendicular to the plane of the directrix, which is the curve serving as the base. In this configuration, the two parallel bases of a bounded right cylinder are directly aligned one above the other, with the axis perpendicular to both bases, resulting in sides that stand straight up relative to the base plane.^[13]^[14] This perpendicular orientation ensures that the lateral surface consists of rectangular strips when unrolled, simplifying geometric analysis in many contexts.^[15] In contrast, an oblique cylinder features rulings that intersect the base plane at an angle other than 90 degrees, causing the lateral sides to slant or "lean" relative to the bases. The bases remain parallel, but their positions are offset, leading to a sheared appearance where the top base is shifted laterally from the bottom one. This obliqueness does not alter the fundamental cylindrical nature but introduces asymmetry in the spatial arrangement.^[16]^[17] Both types of cylinders can be constructed by extruding a fixed curve, the directrix, along a straight-line vector: for a right cylinder, the vector is perpendicular to the plane containing the directrix, producing uniform alignment; for an oblique cylinder, the vector is angled to that plane, resulting in the slanted generators. This extrusion method highlights the cylinder as a ruled surface translated parallel to itself.^[15]^[18] A key implication of this distinction lies in measuring height: in a right cylinder, the height equals the length of the rulings, as they align with the perpendicular distance between bases. In an oblique cylinder, however, the true height is the shortest perpendicular distance between the parallel base planes, which is less than the slant height—the actual length along the slanted rulings—requiring separate consideration for accurate dimensional analysis.^[17]^[19] For a simple example, consider a rectangular base extruded along a vector: if perpendicular to the base plane, it yields a right rectangular cylinder, akin to a standard box standing upright; if extruded at a 45-degree angle, the resulting oblique rectangular cylinder has its top base shifted sideways, resembling a parallelepiped with rectangular faces but slanted lateral edges.^[13]^[16]

Types of Cylindrical Surfaces

Circular cylinders

A circular cylinder is defined as a cylindrical surface generated by translating a circle, known as the directrix, along a straight line called the ruling or generatrix, with the result that all cross-sections perpendicular to the axis are congruent circles of constant radius.^[2] This configuration ensures that the surface maintains a uniform circular profile along its length, distinguishing it from other cylindrical forms.^[20] In a right circular cylinder, the axis is perpendicular to the planes of the circular bases, imparting full rotational symmetry about the central axis, where any rotation around this axis maps the figure onto itself.^[21] The constant radius of the directrix circle allows for isotropic properties in the plane perpendicular to the axis, making it a fundamental shape in classical geometry. An oblique circular cylinder arises when the axis is inclined at an angle to the base planes, as referenced in the general classification of cylinders; in this case, while cross-sections perpendicular to the axis remain circular, projections onto planes parallel to the bases or certain views appear as ellipses due to the shearing effect.^[7]^[22] This variant retains the rotational symmetry and constant radius inherent to the circular directrix but introduces asymmetry in orientation. When considered as a bounded solid, a circular cylinder consists of two parallel circular bases connected by the lateral cylindrical surface, forming a closed three-dimensional figure with uniform circular extremities.^[12] A representative example is a standard beverage can, which approximates a right circular cylinder with its circular top and bottom bases aligned perpendicular to the vertical axis.

Non-circular cylinders

Non-circular cylinders are cylindrical surfaces generated by translating a non-circular curve, known as the directrix, along a straight line path parallel to a fixed direction, producing infinite ruled surfaces that extend indefinitely.^[5] The directrix can be any plane curve, such as an ellipse, hyperbola, parabola, or more general conic or non-conic curve, resulting in surfaces without the rotational symmetry of circular cylinders.^[23] An elliptic cylinder arises when the directrix is an ellipse; in this case, all bounded cross-sections perpendicular to the generating lines remain ellipses, preserving the elliptical profile throughout the surface./12:_Vectors_in_Space/12.06:_Quadric_Surfaces) This configuration yields a surface that is bounded in the plane of the directrix but unbounded along the translation direction.^[5] The hyperbolic cylinder features a hyperbola as its directrix, /12:_Vectors_in_Space/12.06:_Quadric_Surfaces) A parabolic cylinder is defined by a parabolic directrix, leading to cross-sections that are parabolas in planes parallel to the directrix; this form is particularly relevant in optics for parabolic reflectors and in mechanics for analyzing stress distributions.^[24] The surface's unbounded nature in both the directrix plane and the generation direction makes it suitable for modeling open-ended phenomena.^[5] In algebraic geometry, non-circular cylinders correspond to degenerate quadric surfaces, characterized by a quadratic form whose associated matrix has one eigenvalue equal to zero, reducing the rank and allowing the surface to factor into a product of a linear and a quadratic term.^[25] This degeneracy distinguishes them from non-degenerate quadrics like ellipsoids, as cylinders are translationally invariant and extend infinitely in one principal direction without closure.^[26]

Geometric Properties

Cylindric sections

A cylindric section refers to the curve formed by the intersection of a plane with the surface of a cylinder. In general, when the intersecting plane is perpendicular to the rulings (the straight-line generators parallel to the cylinder's axis), the resulting section is a copy of the cylinder's directrix curve, the fixed curve in the base plane through which the rulings pass. For planes at other angles to the rulings, the intersection produces conic sections, limited to ellipses (including circles as a special case) or degenerate forms such as lines, due to the parallel nature of the rulings on the cylindrical surface. For a right circular cylinder, where the directrix is a circle and the rulings are perpendicular to the base plane, the sections vary distinctly by plane orientation. A plane perpendicular to the axis yields a circle identical to the directrix. A plane parallel to the axis intersects the surface in two parallel straight lines, corresponding to two generators. An oblique plane, neither perpendicular nor parallel to the axis, produces an ellipse, with the eccentricity depending on the angle of inclination.^[1]^[27] In the case of an oblique circular cylinder, where the rulings are parallel but slanted relative to the base plane, the sections differ slightly. A plane perpendicular to the rulings still results in a circle, but such planes are not aligned with the base. All other non-parallel intersections yield ellipses, and circles do not appear unless the plane is specifically perpendicular to the rulings. Planes parallel to the rulings produce degenerate sections of one or two parallel lines, similar to the right case. For non-circular cylinders, such as elliptic or parabolic cylinders, the sections follow analogous principles but reflect the shape of the directrix. Perpendicular intersections reproduce scaled or congruent versions of the directrix curve itself. Oblique planes generate conic sections that are stretched or sheared versions of the directrix, typically ellipses for bounded directrices like ellipses, while unbounded ones like parabolas yield parabolic sections under certain angles. The exact form depends on the angle between the plane and the rulings, preserving the conic nature without introducing hyperbolas. Special cases arise when the plane is parallel to the rulings: the intersection consists of straight lines along the generators, either one line if the plane is tangent to the cylinder or two parallel lines if it secants the surface without further intersection. Tangent planes, by definition, touch the cylinder along a single generator, yielding a straight-line section. These degenerate cases highlight the boundary behaviors of cylindric sections.^[27] Historically, the study of cylindric sections contributed to early understandings of conic generation; Euclid noted in his Phaenomena that a plane cutting a cylinder not parallel to its base produces a section similar to a shield, an observation that paralleled emerging ideas on ellipses from conical sections.^[28]

Volume

The volume of a bounded cylinder is given by the formula V = A h, where A is the area of the base (or directrix) and h is the perpendicular height between the two parallel bases.^[29] This formula holds for any cylinder, as the volume depends only on the cross-sectional area and the perpendicular distance, not on the orientation of the generating lines.^[19] For a right circular cylinder, the base is a disk of radius r, so A = \pi r^2, yielding V = \pi r^2 h.^[30] This can be derived geometrically by viewing the cylinder as a prism with a circular base, where the volume is the base area times height, analogous to polygonal prisms.^[30] Alternatively, using integration, consider slicing the cylinder perpendicular to its axis into thin disks of thickness dy; each disk has volume \pi r^2 \, dy, so integrating from 0 to h gives

V = \int_0^h \pi r^2 \, dy = \pi r^2 h.

^[31] In an oblique cylinder, the generating lines are not perpendicular to the bases, but the volume remains V = A h, with h as the perpendicular distance between bases rather than the length along the rulings.^[19] This invariance follows from Cavalieri's principle: cross-sections parallel to the bases have the same area A at corresponding heights, so the volumes match that of a right cylinder with identical base and perpendicular height.^[19] For non-circular cylinders, where the base is a region bounded by a curve (directrix) in the plane, the volume is V = A h, with A the area enclosed by the directrix and h the perpendicular height.^[29] A proof sketch via slicing treats the cylinder as stacked thin plates parallel to the base, each of area A and thickness dh; the total volume is then \int_0^h A \, dh = A h, applicable by Cavalieri's principle to both right and oblique cases.^[32] The volume is measured in cubic units, such as cubic meters for engineering applications like calculating the capacity of a cylindrical tank, where V represents the enclosed space.^[30]

Surface area

The lateral surface area of a bounded cylinder is given by the product of the perimeter of the directrix curve and the length of the generator line segment connecting corresponding points on the two bases.^[33] For a right cylinder, where the generators are perpendicular to the bases, this length equals the perpendicular height h between the bases. In an oblique cylinder, the generators are slanted, so the length is the slant height l, which exceeds h. For a right circular cylinder of radius r, the perimeter of the directrix (a circle) is $2\pi r, yielding a lateral surface area of $2\pi r h.^[1] The total surface area, including the two circular bases each of area \pi r^2, is then $2\pi r h + 2\pi r^2.^[1] This derivation arises from unrolling the lateral surface into a rectangle of width $2\pi r and height h, whose area is the product of these dimensions.^[34] In an oblique circular cylinder, the lateral surface area is $2\pi r l, where the slant height l = h / \cos \theta and \theta is the obliqueness angle between the generator and the perpendicular axis.^[16] The total surface area adds the areas of the two bases, $2\pi r l + 2\pi r^2. The unrolling derivation holds similarly, with the rectangle's height now l instead of h. For non-circular cylinders, the lateral surface area is the length of the directrix curve times the generator length ( h for right, l for oblique).^[33] For instance, in manufacturing a cylindrical can, the lateral surface area calculates the material or paint required for the curved sides, separate from the top and bottom ends.^[19]

Cylindrical shells

A cylindrical shell, or hollow cylinder, is the three-dimensional region bounded between two coaxial cylindrical surfaces sharing the same axis, with the inner cylinder having radius r_1 and the outer cylinder having radius r_2 > r_1, extending along a height h. This structure forms a tube-like solid, often modeled as the difference between two solid cylinders.^[35] For a right circular cylindrical shell, the volume is derived by subtracting the volume of the inner solid cylinder from that of the outer, yielding V = \pi (r_2^2 - r_1^2) h. The total surface area consists of the inner lateral surface $2\pi r_1 h, the outer lateral surface $2\pi r_2 h, and two annular end faces, each with area \pi (r_2^2 - r_1^2). Thus, the total surface area is $2\pi h (r_1 + r_2) + 2\pi (r_2^2 - r_1^2).^[36]^[37] In the case of an oblique cylindrical shell, where the bases are parallel but offset along the axis, the volume formula adjusts to use the perpendicular height h between the bases, maintaining V = \pi (r_2^2 - r_1^2) h by Cavalieri's principle, which equates volumes of solids with equal cross-sectional areas at corresponding heights. The surface areas follow similar distinctions, with lateral areas based on the generating lines' lengths adjusted for obliquity, though the perpendicular height governs the core geometric computations. For thin cylindrical shells, where the wall thickness t = r_2 - r_1 is small relative to the radii (typically t / r_1 < 0.1), approximations simplify calculations; the total lateral surface area is often estimated using the mean radius r_m = (r_1 + r_2)/2, giving $2\pi r_m h, which provides sufficient accuracy for many engineering analyses while focusing on geometric essentials. In engineering contexts, such as pipes or pressure vessels, these properties inform designs by enabling precise volume and area computations essential for material efficiency and load-bearing assessments, though the primary emphasis remains on the underlying geometry.

Historical Development

Archimedes' contributions

Archimedes of Syracuse composed his seminal treatise On the Sphere and Cylinder around 225 BCE, a two-volume work in which he analyzed the properties of cylinders, defined as solids bounded by two parallel circles and the surface generated by straight lines connecting their circumferences, establishing precise geometric relationships between cylinders and spheres. This text represents one of the earliest systematic investigations into the properties of cylindrical solids, focusing on their volumes and surfaces through rigorous geometric propositions.^[38] A central achievement in the treatise is Archimedes' demonstration that the volume of a sphere is two-thirds the volume of the circumscribing cylinder, where the cylinder's height equals the sphere's diameter. He further proved that the surface area of the sphere equals the curved (lateral) surface area of this same circumscribed cylinder, a result derived without explicit use of transcendental constants but through proportional comparisons. These theorems underscore the cylinder's role as a bounding figure for the sphere, providing foundational insights into three-dimensional geometry.^[38] Archimedes employed the method of exhaustion—a technique of successively inscribing and circumscribing polygonal approximations to bound the figures and reduce discrepancies to negligible limits—as the primary tool for his proofs, serving as a precursor to the integral calculus developed millennia later. Complementing this, he utilized a mechanical heuristic involving balancing levers to investigate volumes intuitively, conceptualizing cross-sections of the solids as weights equilibrated on a beam to reveal proportional relationships before formal verification.^[39]^[38] In detailing the cylinder-sphere volume ratio, Archimedes offered a comprehensive proof showing that the sphere's volume is two-thirds that of the circumscribed cylinder with base radius matching the sphere's and height equal to the diameter, equivalent in modern terms to the sphere's volume being \frac{4}{3} \pi r^3 and the cylinder's $2 \pi r^3. This relation not only quantifies their interdependence but also illustrates Archimedes' mastery of spatial proportions. The work was produced in Syracuse amid the Hellenistic patronage of King Hiero II, reflecting the vibrant intellectual environment of the Sicilian court where Archimedes served as an engineer and scholar. Although many of his manuscripts faced loss during the Roman era and medieval disruptions, On the Sphere and Cylinder endured through careful copying in Byzantine scriptoria and subsequent translations into Arabic and Latin, facilitating its rediscovery and enduring legacy in mathematical literature.^[40]

Post-ancient advancements

During the Renaissance, interest in classical geometry revived through scholarly commentaries on Euclid's Elements and Archimedes' works, including his foundational treatise On the Sphere and Cylinder, which explored volumetric relationships between these solids.^[41] This revival was facilitated by the translation and annotation of ancient texts, such as Eutocius's commentaries on Archimedes, preserved in Renaissance manuscripts that emphasized practical geometric constructions.^[41] In artistic applications, Leon Battista Alberti's De pictura (1435) advanced the representation of cylindrical forms in perspective drawing, instructing artists on rendering circular bases as ellipses to achieve realistic depth for objects like columns and vessels.^[42] In the 17th century, René Descartes's La Géométrie (1637) marked a pivotal shift by integrating algebra with geometry, parametrizing cylinders as quadric surfaces through equations that described their generating lines parallel to a fixed direction.^[43] This analytic approach allowed for systematic classification of surfaces, treating cylinders as degenerate quadrics bounded by conic sections.^[44] The 18th and 19th centuries saw further classification of cylindrical surfaces within differential geometry, with Carl Friedrich Gauss's Disquisitiones generales circa superficies curvas (1827) identifying them as ruled surfaces with zero Gaussian curvature, distinguishing them from more complex curved forms.^[45] Concurrently, Gaspard Monge's descriptive geometry, developed around 1795 for military and engineering purposes, provided methods to project and intersect cylindrical surfaces, enabling accurate technical drawings of machine parts like pipes and shafts.^[46] Oblique cylinders, where the generating lines are not perpendicular to the base, were formalized in solid mensuration texts, notably Adrien-Marie Legendre's Éléments de géométrie (1794), which included derivations for their volumes and areas to support practical computations.^[47] As geometry transitioned toward modern frameworks in the 19th century, cylinders illustrated key concepts in vector calculus and differential geometry; their straight rulings served as geodesics, representing the shortest paths on the surface when unrolled into a plane.^[48] In engineering, cylindrical forms gained prominence in steam engine design, where precise geometric analysis of cylinder bores and pistons optimized pressure-volume relations, as exemplified in James Watt's improvements and subsequent industrial applications.^[49]

Mathematical Contexts

Cylindrical coordinates

Cylindrical coordinates provide a natural framework for describing points in three-dimensional space, particularly those exhibiting rotational symmetry around an axis. This system extends the two-dimensional polar coordinates by incorporating a vertical dimension, specifying each point using three parameters: the radial distance \rho \geq 0 from the reference z-axis, the azimuthal angle \phi measured from the positive x-axis in the xy-plane (typically $0 \leq \phi < 2\pi), and the axial coordinate z along the axis.^[50] The coordinates originate from the projection of the point onto the xy-plane, where \rho and \phi define the polar position, with z giving the height.^[51] The transformation between cylindrical and Cartesian coordinates is given by the equations

x = \rho \cos \phi, \quad y = \rho \sin \phi, \quad z = z,

allowing straightforward conversion for computations involving distances or vectors.^[52] In this system, the equation of a right circular cylinder aligned with the z-axis and radius a simplifies to \rho = a, highlighting the coordinate system's alignment with the geometry of such surfaces. For oblique cylinders, where the generators are not perpendicular to the base, or non-circular cylinders with arbitrary cross-sections, the equation takes a more general form, such as f(\rho, \phi) = c for the cross-sectional curve in the \rho-\phi plane, though standard cylindrical coordinates assume circular symmetry and parallel generators along z./12%3A_Vectors_in_Space/12.07%3A_Cylindrical_and_Spherical_Coordinates) A key feature for integration in cylindrical coordinates is the volume element, derived from the Jacobian of the transformation, which is dV = \rho \, d\rho \, d\phi \, dz. This factor of \rho accounts for the varying "stretch" in the radial direction and is essential for setting up triple integrals over cylindrical regions, such as deriving volumes or computing physical quantities like mass or charge distribution.^[51] These coordinates offer significant advantages in problems with cylindrical symmetry, such as those involving flux through cylindrical boundaries or electromagnetic fields around wires, by reducing complex integrals to separable forms and simplifying boundary conditions. For instance, the lateral surface of a right circular cylinder of radius a can be parametrized as

\mathbf{r}(\phi, z) = (a \cos \phi, a \sin \phi, z),

with \phi \in [0, 2\pi] and z spanning the height, facilitating calculations of surface area or vector fields tangent to the surface. Cylindrical coordinates are especially suited to the rotational symmetry inherent in circular cylinders.^[52]^[53]

Projective geometry

In projective geometry, a cylinder is defined as a quadric surface in three-dimensional projective space \mathbb{P}^3, generated by a one-parameter family of straight lines (known as rulings) that lie on the surface and pass through points of a conic curve (the directrix) in a plane, with the rulings being parallel in the affine patch, equivalent to a cone whose apex is at infinity.^[24] This structure makes the cylinder a ruled quadric, preserving the incidence properties under projective transformations, where lines map to lines and conics to conics.^[25] The key properties of cylinders in projective space include the invariance of their rulings under projection: any projective transformation maps the rulings to straight lines, maintaining the ruled nature of the surface. Conic cylinders, those with a conic directrix, project onto conics in the projective plane, as the envelope formed by the projected rulings bounds a conic region. For instance, the orthogonal projection of a circular cylinder onto a plane yields an elliptical silhouette, while perspective projection from a viewpoint produces a general conic envelope, demonstrating how cylinders unify various conic types through projection.^[54] Cylinders represent degenerate quadrics of rank 3, arising as limits of non-degenerate quadrics such as the hyperboloid of one sheet, where one family of rulings becomes parallel (sent to infinity), or as a pair of distinct planes in further degeneration when the directrix collapses.^[25] In applications, particularly in computer vision, the projective properties of cylinders facilitate shape reconstruction and relative pose estimation from multiple views, leveraging the conic silhouettes and parallel rulings for recovering 3D structure without metric assumptions.^[55] In algebraic geometry, modern studies explore projective invariants of cylinders embedded in smooth minimal geometrically rational surfaces, determining existence conditions over perfect fields, such as the presence of rational curves admitting cylindrical structures.^[56]

Relation to prisms

A prism is a polyhedron consisting of two parallel, congruent polygonal bases connected by rectangular or parallelogram lateral faces. In a right prism, the lateral faces are perpendicular to the bases, whereas in an oblique prism, they form an angle, mirroring the distinction between right and oblique cylinders where the generating lines are perpendicular or slanted relative to the bases.^[18] Cylinders can be conceptualized as the limiting case of prisms with regular polygonal bases as the number of sides approaches infinity; for instance, a prism with an n-sided regular polygonal base inscribed in a circle of radius r approximates a right circular cylinder.^[57]^[58] As n \to \infty, the area of the polygonal base converges to \pi r^2, and the perimeter to $2\pi r, such that the prism's volume formula V = A_b h (where A_b is the base area and h the height) approaches the cylinder's V = \pi r^2 h, while the lateral surface area S = P h (with P the base perimeter) approaches $2\pi r h.^[59]^[60] Despite these convergences, prisms and cylinders differ fundamentally: prisms possess a finite number of flat polygonal faces and belong to polyhedral geometry, whereas cylinders feature a continuous smooth lateral surface generated by a straight line along a curved directrix.^[61] For example, a triangular prism, with its flat triangular bases and three rectangular lateral faces, approximates a cylindrical wedge—a sector of a cylinder bounded by two radial planes—but retains discrete facets rather than the wedge's curved boundary.^[62]^[59]

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We first consider solids whose slices are thin cylinders. Recall that the volume of a cylinder is given by . V = π r 2 h . Example 6.18. Find the volume of the ...
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Volume Patterns for Pyramids - Brown Math
Once we observe this relationship, we can express it in formula: the volume of a cylinder or prism is the area of the base multiplied by the height, and the ...Missing: geometric | Show results with:geometric
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13.2 Cylinders - The Geometry Center
When =90° we have a right cylinder, and h=l. For a right cylinder, the lateral area between P and P' is hs, where s is the length (circumference) of C. r h. x ...
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Calculus II - Surface Area - Pauls Online Math Notes
Nov 16, 2022 · We can derive a formula for the surface area much as we derived the formula for arc length. We'll start by dividing the interval into n n equal ...
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Volume of Hollow Cylinder Formula - BYJU'S
If “R” is the outer radius and “r” is the inner radius and “h” is the height, then the volume of the hollow cylinder formula is V = π (R2 -r2)h cubic units. Q4 ...
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[PDF] READING ASSIGNMENT 05 NAME: SOLUTIONS
(1) What is the volume of a hollow cylinder of inner radius r, outer radius R, and height h? R r h. ANSWER: π(R2 − r2)h = πh(R + r)∆r, where ∆r = (R − r) ...
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Area of Hollow Cylinder - GeeksforGeeks
Jul 23, 2025 · Here are formulas for other areas of a hollow cylinder: Lateral Surface Area (LSA): 2πh (R + r) Cross-Sectional Area: π(R2 − r2)
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Oblique Cylinder Calculator and Formula - Redcrab Home
Definition: Circular cylinder with lateral surface inclined to the base ... Surface Area Calculation. Lateral area and total surface area: L=2πra=2π⋅3 ...
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The washer method - Ximera - The Ohio State University
To find the volume of the hollow cylinder, recall. The outer cylinder has radius and its volume is , while the volume of the inner cylinder has radius , so its ...
[41]
Simple Thin Pressure Vessels | Engineering Library
May 8, 2019 · mean radius. r, = radius to the inside of the skin of stiffened cylinder. r ... Figure 8-2 shows a thin cylindrical shell (R/t > 10). Thin ...Missing: approximation | Show results with:approximation<|control11|><|separator|>
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Archimedes - Biography - MacTutor - University of St Andrews
In the first book of On the sphere and cylinder Archimedes shows that the surface of a sphere is four times that of a great circle, he finds the area of any ...
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The Method of Archimedes - AMS :: Feature Column from the AMS
Archimedes determined the ratio of the volume of a sphere to the volume of the circumscribed cylinder. The actual construction involves the cylinder ...Missing: sources | Show results with:sources
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Between myth and mathematics: the vicissitudes of Archimedes and ...
Aug 16, 2013 · In what follows we will discuss this complex history: from third century B.C. Syracuse ... Archimedes' On the Sphere and Cylinder. We might also ...
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The Vatican Library & Renaissance Culture > Mathematics
Shown here is a part of Eutocius's commentary on Archimedes' On the Sphere and the Cylinder, in which he reviews solutions to the classical problem of the ...
[46]
Alberti, On Painting (excerpts)
In 1435 he completed in Latin his groundbreaking treatise De pictura, and the next year he dedicated an Italian version of the text to Brunelleschi. He presents ...Missing: cylinder | Show results with:cylinder
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Descartes' Mathematics - Stanford Encyclopedia of Philosophy
Nov 28, 2011 · To speak of René Descartes' contributions to the history of mathematics is to speak of his La Géométrie (1637), a short tract included with ...
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[PDF] The geometry of Renâe Descartes
De La Géométrie. Comment 0» peut faire vn verre autant connexe ou concatig en l*vne de fes fuperficieStCju on voudra, ^uirajfemble a vn point donné tout les.
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[PDF] General investigations of curved surfaces of 1827 and 1825 ...
J. Page 9. INTRODUCTION. In 1827 Gauss presented to the Royal Society of Gottingen his important ...
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Gaspard Monge and descriptive geometry - jstor
3 Given a cylinder or conic, to find the tangent at a particular point. 4 To describe the intersections of surfaces ?cylinders, conies, etc. 5 To investigate ...
[51]
Elements of geometry : Legendre, A. M. (Adrien Marie), 1752-1833
Oct 19, 2009 · 78 RPMs and Cylinder Recordings. Top. Audio Books & Poetry · Computers ... Elements of geometry. by: Legendre, A. M. (Adrien Marie), 1752-1833.Missing: oblique cylinders solid mensuration 1794
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[PDF] DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
Apr 26, 2021 · This course covers curves, local theory of surfaces, further topics on surfaces, and a review of linear algebra and calculus.
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The History of the Steam Engine (1890) - Vintage Machinery Wiki
Mar 15, 2012 · During the first half of the nineteenth century progress in steam engineering was very largely in the direction of the application of the steam ...
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Cylindrical Coordinates -- from Wolfram MathWorld
Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis.
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Calculus III - Cylindrical Coordinates - Pauls Online Math Notes
Nov 16, 2022 · Cylindrical coordinates extend polar coordinates into 3D by adding a z coordinate. x=rcosθ, y=rsinθ, z=z are the conversions.
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2.7 Cylindrical and Spherical Coordinates - Calculus Volume 3
Mar 30, 2016 · As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a ...<|separator|>
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Calculus III - Parametric Surfaces - Pauls Online Math Notes
Mar 25, 2024 · In this section we will take a look at the basics of representing a surface with parametric equations.
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[PDF] Relative pose from cylinder silhouettes - CVF Open Access
Abstract. In this paper we propose minimal solvers for relative pose estimation for two views of the projected silhouettes of two 3D cylin-.
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The Multi-view Geometry of Parallel Cylinders - AI Lund
In this paper we study structure from motion problems for parallel cylinders. Using sparse keypoint correspondences is an efficient (and standard) way to ...<|control11|><|separator|>
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[2204.08864] Notes on cylinders in smooth projective surfaces - arXiv
Apr 19, 2022 · In this article, we determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field.
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Relation of a cylinder to a prism - Math Open Reference
A cylinder can be thought of as a right regular prism with an infinite number of sides.
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Prisms, Cylinders & Limits - MathBitsNotebook(Geo)
As the number of lateral faces (sides) increases indefinitely (approaches infinity), the two solids will appear to blend into one solid (the cylinder). bullet ...
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Prism vs Cylinder: Key Differences, Formulas & Examples - Vedantu
Rating 4.2 (373,000) For a prism, the volume is V = (Area of the polygonal base) × h. · For a cylinder, the volume is V = (Area of the circular base) × h, which simplifies to V = ...
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9.03 Prisms and cylinders - Textbooks :: Mathspace
A cylinder with the height drawn from the center of the top circle to the center. When a prism or cylinder becomes oblique, its volume will remain the same ...
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Are right circular cylinders prismatic? - Mathematics Stack Exchange
May 21, 2018 · 1. Cylinder isn't a prism because it has curved sides. Sonal_sqrt · Polygons may approximate a circle, yet a circle is not a polygon. Likewise, ...
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Wedge -- from Wolfram MathWorld
"Wedge" refers a right triangular prism turned so that it rests on one of its lateral rectangular faces.