Cylinder

A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. Cylinders are found all around us, from the cans we use in the kitchen to the tall towers that rise in cities.

Cylinders can also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. This shift in meaning—solid versus surface—has created some confusion with terminology. To make things clear, people may refer to solid cylinders and cylindrical surfaces. In many books and articles, the simple term "cylinder" could mean either of these, or even a more specific object, the right circular cylinder.

The study of cylinders helps us understand shapes and spaces better, and it plays an important role in many fields, from engineering to art.

Types

A cylindrical surface is made up of all the points on lines that are parallel to a given line and pass through a fixed curve in a plane. These lines are called elements of the cylindrical surface.

A solid bounded by a cylindrical surface and two parallel planes is called a solid cylinder. The line segments between the two planes are called elements of the cylinder, and they all have the same length. The flat circular ends of the cylinder are called its bases. If the elements are perpendicular to the bases, the cylinder is a right cylinder; otherwise, it is an oblique cylinder. When the bases are circles, the cylinder is a circular cylinder.

The height of a cylinder is the distance between its two bases. A cylinder made by rotating a line segment around a fixed line parallel to it is called a cylinder of revolution, and it is always a right circular cylinder.

Right circular cylinders

Main article: Right circular cylinder

When people talk about a cylinder, they often mean a right circular cylinder—a solid with circular ends perpendicular to its axis. The flat ends without the sides are called an open cylinder. We have known the formulas for the surface area and volume of right circular cylinders since ancient times.

A right circular cylinder can also be created by rotating a rectangle around one of its sides. These cylinders are used in a math method called the "disk method" to find the volumes of solids formed by rotation. A needle cylinder is tall and thin, while a disk cylinder is short and wide.

Properties

A cylinder is a 3D shape with circular ends and a curved side. When a plane cuts through a cylinder, the shape of the cut depends on the angle of the plane. If the plane cuts straight through the center, the shape is a circle. If it cuts at an angle, the shape can be an ellipse or other curved lines.

We can find the amount of space inside a cylinder, called its volume. For a cylinder with a circular end that has a radius r and a height h, the volume is calculated by multiplying the area of the circle (π × r²) by the height (h). This gives us the formula V = πr²h. This same formula works for cylinders that stand straight up and for those that are tilted.

Cylindrical surfaces

In geometry and topology, a cylindrical surface is a special kind of surface made up of lines that run parallel to each other and pass through a fixed curve in a plane. These surfaces are sometimes called generalized cylinders.

Cylinders can have different shapes depending on the curve used. For example, if the curve is an ellipse, parabola, or hyperbola, the cylinder is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder, respectively. These are special types of surfaces known as degenerate quadric surfaces.

Projective geometry

In projective geometry, a cylinder is like a special type of cone where the pointy tip, called the apex, is very far away, almost out of sight. This idea helps us understand different kinds of shapes called degenerate conics, which can include cylindrical conics.

Prisms

A solid circular cylinder is like a special kind of prism when you imagine a many-sided shape that becomes more and more rounded. Because of this, many old geometry books talk about prisms and cylinders together. We can find the size and space inside a cylinder using ideas from prisms, by thinking about shapes with many sides that become perfectly round.

Circular cylinders are special because their round shape makes them easier to study using simple math. Cylinders and prisms also use similar names — for example, a prism with bases that aren’t aligned is called a truncated prism, and a cylinder with bases that aren’t aligned is called a truncated cylinder. From another angle, a cylinder can also be thought of as linked to a bicone.