Cylinder

A cylinder is a simple 3D shape, one of the most basic curved shapes. In elementary geometry, it is like a prism with a circle as its base. We see cylinders everywhere, such as in the cans we use in the kitchen and the tall towers in cities.

Cylinders can also be thought of as a curved surface in more advanced geometry and topology. This can sometimes make the meaning a little confusing. To help clear this up, people talk about solid cylinders and cylindrical surfaces. In many books and articles, the word "cylinder" might mean either of these, or sometimes a special type called the right circular cylinder.

Learning about cylinders helps us understand shapes and spaces better. They are important in many areas, from engineering to art.

Types

A cylindrical surface is made of 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 flat surface cuts through a cylinder, the shape of the cut depends on the angle. If the surface 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 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 found by multiplying the area of the circle (π × r²) by the height (h). This gives us the formula V = πr²h. This formula works for cylinders that stand straight and for those that are tilted.

Cylindrical surfaces

In geometry, a cylindrical surface is a special kind of surface. It is made of lines that run parallel to each other and pass through a fixed curve in a plane.

Cylinders can have different shapes based 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. These are special types of surfaces called degenerate quadric surfaces.

Projective geometry

In projective geometry, a cylinder is a special shape. It is like a cone where the tip, called the apex, is very far away. This idea helps us learn about different shapes called degenerate conics.

Prisms

A solid circular cylinder is a special kind of shape. It is like a prism that has been rounded more and more until it becomes perfectly round. 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. We do this by thinking about shapes with many sides that become perfectly round.

Circular cylinders are special because their round shape makes them easier to study with 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.