The main parts of a cylinder are the faces, edges, and vertices. The cylinder is a three-dimensional figure, but it is not considered a polyhedron, so the parts of the cylinder are considered a little differently than the parts of polyhedra. However, we can define the parts of cylinders using a structure similar to that of polyhedra, specifically, we use a CW complex structure. When using CW complex, we can consider faces and edges that have curvature. Therefore, using this structure, we can distinguish 3 faces, 3 edges, and 2 vertices on the cylinder.
Here, we are going to learn about the parts of a cylinder in more detail. We will use diagrams to illustrate the concepts.
Faces found of a cylinder
The faces of cylinders are different from the faces of a polyhedron, in the sense that not all the faces of a cylinder are planar surfaces. In a cylinder, we have two circular faces and one face that has a curved surface.
The two circular faces form the bases of the cylinder and are generally parallel to each other. The third face can be stretched to form a rectangle as shown in the diagram:
By adding the areas of the cylinder’s faces, we can calculate its surface area. Circular faces have an area that is calculated with the expression πr², where r is the length of the radius. Since we have two circular bases, their area is equal to 2πr².
The area of the lateral surface is found by recognizing that, if we stretch it, we form a rectangle that has a length equal to the circumference of the cylinder bases and a height equal to the height of the cylinder.
Therefore, the area of the third face is 2πrh. This means that the surface area of the cylinder is 2πr² + 2πrh.
Vertices of a cylinder
For polyhedra in general, we define vertices as the points where two or more edges meet. However, in the case of cylinders, this definition is slightly different. Here, we consider the CW complex structure, as mentioned in the introduction.
Using this structure, we can consider faces that have curved surfaces and edges that have curvature. Therefore, we define the vertices of the cylinder as the points that are located in each circle.
This can be interpreted as the point where we start the circle and where we end a complete turn. Therefore, we have two vertices in a cylinder.
Edges of a cylinder
For polyhedra in general, edges are defined as line segments that join two vertices. However, similar to vertices, we define the edges of a cylinder using the CW complex structure. This allows us to have edges that have curves.
Therefore, the cylinder has two curved edges, which are the circumference of the circles at the bases, and an edge that connects the two vertices and is located along the lateral face. Therefore, a cylinder has a total of 3 edges.
We can conclude that cylinders have 3 faces, 2 vertices, and 3 edges. This satisfies the Euler characteristic, which is a definite number that allows us to describe the structure of polyhedra or topological spaces.
This means that the Euler characteristic of a cylinder is 2-3 + 3 = 2. This agrees with the Euler characteristic of 2 of a sphere, which is correct since a cylinder is homotopically equivalent to a sphere.
Interested in learning more about cylinders? Take a look at these pages: