Cylinder

A cylinder is a solid shape.

It has two flat ends. These ends are called bases. The bases are the same size.

A tall cylinder looks like a needle. A short cylinder looks like a disk. You can also have a hollow cylinder. This looks like a tube.

A long time ago, a man named Archimedes studied them. He found how they work with spheres. He was very proud of this work.

A cylinder is a basic 3D shape.

It can be a solid object or just a surface. Most people think of a solid cylinder with two circular ends. These flat ends are called bases. The two bases are exactly the same size.

The distance between the bases is called the height. If the sides go straight up, it is a right cylinder. If they lean, it is an oblique cylinder.

There are many types of cylinders. A circular cylinder has round bases. An elliptic cylinder has bases shaped like an ellipse. You can also have a parabolic cylinder.

A hollow cylinder looks like a tube.

A long cylinder looks like a needle. A wide one looks like a disk.

A famous thinker named Archimedes studied these shapes. He found a special link between a sphere and a cylinder. He found that a sphere has two-thirds the volume of a cylinder that fits it perfectly.

This was a discovery he was very proud of. He even asked for these shapes to be put on his tomb.

A cylinder is a fundamental shape in our three-dimensional world.

You might think of it as a solid object, like a can of soup. However, mathematicians also see it as a surface that stretches through space. In geometry, a cylinder is often seen as a prism with a circular base. You can imagine a line moving parallel to itself along a curved path. This movement traces out a cylindrical surface. If the sides of the shape stand straight up, it is called a right cylinder. If the sides lean to one side, it is an oblique cylinder.

There are many different ways to build a cylinder. A circular cylinder is the most common type because its bases are perfect circles. If the bases are shaped like an ellipse, it is an elliptic cylinder.

You can even find parabolic cylinders or hyperbolic cylinders. Some cylinders are very tall and thin, like a needle. Others are short and wide, like a heavy disk. If you take a solid cylinder and remove the middle, you get a hollow cylinder.

This shape looks like a tube or a shell. You can calculate its volume by subtracting the inner empty space from the outer shape.

When a flat plane cuts through a cylinder, it creates a special shape called a cylindric section.

The shape of this section depends on how the plane hits the cylinder. If the plane cuts straight across a right cylinder, the section is a circle. If the plane cuts at an angle, the section becomes an ellipse.

If the plane cuts through the sides vertically, the section might look like a rectangle. In more advanced math, these sections can even be parts of a parabola. These different slices help us understand how the shape occupies space.

Math allows us to measure these shapes with great precision. To find the volume of a circular cylinder, you multiply the area of the base by the height.

The surface area is the total area of the top, the bottom, and the curved side. The curved part is often called the lateral area. For a right circular cylinder, the surface area includes the two circular bases and this side area. If the cylinder is hollow, it might not have any bases at all. This means it would only have a lateral area.

One of the most famous stories in math involves the thinker Archimedes. He studied the relationship between a sphere and a cylinder.

He discovered that a sphere has two-thirds the volume of a cylinder that fits it perfectly. He also found the surface area of a sphere using this same connection. This discovery was so important to him that he requested these shapes be placed on his tomb. His work shows how different shapes are linked by beautiful patterns. Even today, we use these ideas to understand the world around us.

A cylinder is a fundamental three-dimensional shape used in geometry and topology. In elementary geometry, it is often defined as a prism with a circular base. However, in modern mathematics, the term can describe an infinite curvilinear surface. This distinction creates some ambiguity in technical literature. To be precise, mathematicians distinguish between solid cylinders and cylindrical surfaces. A cylinder can be thought of as a collection of points on lines that are parallel to a given line. These lines pass through a fixed plane curve.

To understand how a cylindrical surface is formed, we can look at its mechanism through kinematics. We begin with a plane curve known as the directrix. A line called the generatrix moves parallel to itself while always passing through this directrix. The path traced by this moving line creates the cylindrical surface. Every specific position of the generatrix is called an element of the surface. When we bound this surface with two parallel planes, we create a solid cylinder. The line segments between these planes are also called elements, and all elements in a cylinder have equal lengths.

Cylinders are classified by their orientation and the shape of their bases. If the elements are perpendicular to the planes containing the bases, it is a right cylinder. If the elements lean at an angle, it is called an oblique cylinder. When the bases are disks, the shape is a circular cylinder. A cylinder of revolution is a specific type of right circular cylinder. This shape is generated by rotating a line segment around a fixed axis. The line that the segment revolves around is the axis of the cylinder.

Beyond circles, cylinders can take many other forms based on their cross-sections. If the right section of a cylinder is an ellipse, it is an elliptic cylinder. If the section is a parabola, it is a parabolic cylinder. If the section is a hyperbola, it is a hyperbolic cylinder. These are known as degenerate quadric surfaces. In projective geometry, a cylinder is viewed as a cone whose apex lies at the plane at infinity. This perspective helps mathematicians understand how shapes behave in different geometric systems.

When a plane intersects a cylinder, the resulting shape is called a cylindric section. The geometry of this section depends on the angle of the cut. A plane that is tangent to the cylinder meets it at a single element. If a plane cuts through a right circular cylinder at an angle, the section is an ellipse. If the plane contains two elements of the cylinder, the section is a parallelogram. In a right cylinder, this specific section is a rectangle.

Mathematics provides exact formulas to calculate the properties of these shapes. For a circular cylinder, the volume is the product of the base area and the height. This remains true for both right and oblique cylinders. The surface area of a right circular cylinder consists of three parts: the top base, the bottom base, and the lateral area. The lateral area is the area of the curved side. For a hollow cylinder, or cylindrical shell, the volume is found by subtracting the inner cylinder's volume from the outer cylinder's volume.

One of the most significant historical discoveries regarding cylinders was made by Archimedes. He studied the relationship between a sphere and its circumscribed right circular cylinder. A circumscribed cylinder is one that fits the sphere perfectly, sharing the same height and diameter. Archimedes discovered that the sphere has two-thirds the volume of this cylinder. He also found that the sphere's surface area is two-thirds the total surface area of the cylinder.

This discovery was so meaningful to him that he requested these shapes be placed on his tomb. His work established a deep connection between different geometric solids.