The area of the top and bottom bases is the same, and is called the base area, B. 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: 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:Į = cos α, Surface area Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle. 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. 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. The right sections are circles and all other planes intersect the cylindrical surface in an ellipse. A plane is tangent to the cylinder if it meets the cylinder in a single element. First, planes that intersect a base in at most one point. The line that the segment is revolved about is called the axis of the cylinder and it passes through the centers of the two bases.Ĭylindric sections of a right circular cylinderįor a right circular cylinder, there are several ways in which planes can meet a cylinder. The height of a cylinder of revolution is the length of the generating line segment. A cylinder of revolution is a right circular cylinder. The cylinder obtained by rotating a line segment about a fixed line that it is parallel to is a cylinder of revolution. The height (or altitude) of a cylinder is the perpendicular distance between its bases. In some elementary treatments, a cylinder always means a circular cylinder. If the bases are disks (regions whose boundary is a circle) the cylinder is called a circular cylinder. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a right cylinder, otherwise it is called an oblique cylinder. The two bases of a cylinder are congruent figures. The region bounded by the cylindrical surface in either of the parallel planes is called a base of the cylinder. All the elements of a cylinder have equal lengths. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder. Any particular position of the generatrix is an element of the cylindrical surface.Ī solid bounded by a cylindrical surface and two parallel planes is called a (solid) cylinder. From a kinematics point of view, given a plane curve, called the directrix, a cylindrical surface is that surface traced out by a line, called the generatrix, not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any line in this family of parallel lines is called an element of the cylindrical surface. The definitions and results in this section are taken from the 1913 text Plane and Solid Geometry by George Wentworth and David Eugene Smith ( Wentworth & Smith 1913).Ī cylindrical surface is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the right circular cylinder. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. The shift in the basic meaning-solid versus surface (as in ball and sphere)-has created some ambiguity with terminology. In elementary geometry, it is considered a prism with a circle as its base.Ī cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. A circular right cylinder of height h and diameter d=2 rĪ cylinder (from Ancient Greek κύλινδρος ( kúlindros) 'roller, tumbler') has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. A line can be referred to by two points that lie on it (e.g. The word line may also refer to a line segment in everyday life that has two points to denote its ends ( endpoints). Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. In geometry, a line is an infinitely long object with no width, depth, or curvature.
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