Cylindric sections 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. Such a cylindric section of a right cylinder is a
rectangle. A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a ''''. 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. 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. 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 of the cylindric section and
semi-major axis of the cylindric section depend on the radius of the cylinder and the angle between the secant plane and cylinder axis, in the following way: \begin{align} e &= \cos\alpha, \\[1ex] a &= \frac{r}{\sin\alpha}. \end{align}
Volume If the base of a circular cylinder has a
radius and the cylinder has height , then its
volume is given by V = \pi r^2h This formula holds whether or not the cylinder is a right cylinder. This formula may be established by using
Cavalieri's principle. 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 , semi-minor axis and height has a volume , where is the area of the base ellipse (= ). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive -axis and the area of each elliptic cross-section, thus: V = \int_0^h A(x) dx = \int_0^h \pi ab dx = \pi ab \int_0^h dx = \pi a b h. Using
cylindrical coordinates, the volume of a right circular cylinder can be calculated by integration \begin{align} V &= \int_0^h \int_0^{2\pi} \int_0^r s \,\, ds \, d\phi \, dz \\[5mu] &= \pi\,r^2\,h. \end{align}
Surface area Having radius and altitude (height) , 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: • the area of the bottom base: • the area of the side: The area of the top and bottom bases is the same, and is called the
base area, . The area of the side is known as the '''', . An
open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area) L = 2 \pi r h 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 = 2\pi rh + 2\pi r^2 = 2 \pi r (h + r) = \pi d (r + h) where is the
diameter of the circular top or bottom. For a given volume, the right circular cylinder with the smallest surface area has . Equivalently, for a given surface area, the right circular cylinder with the largest volume has , that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle). The lateral area, , of a circular cylinder, which need not be a right cylinder, is more generally given by L = e \times p, where is the length of an element and is the perimeter of a right section of the cylinder. This produces the previous formula for lateral area when the cylinder is a right circular cylinder.
Right circular hollow cylinder (cylindrical shell) A
right circular hollow cylinder (or '''') 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 , internal radius , and external radius . The volume is given by subtracting the volume of the inner imaginary cylinder (i.e. hollow space) from the volume of the outer cylinder: V = \pi \left( R ^2 - r ^2 \right) h = 2 \pi \left ( \frac{R + r}{2} \right) h (R - r). Thus, the volume of a cylindrical shell equals thickness. The surface area, including the top and bottom, is given by A = 2 \pi \left( R + r \right) h + 2 \pi \left( R^2 - r^2 \right). Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.
On the Sphere and Cylinder In the treatise by this name, written ,
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 that of the circumscribed cylinder and a surface area 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 is . The surface area of this sphere is . A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request. ==Cylindrical surfaces==