Haw many faces does a cylinder have?

Updated: 4/28/2022
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12y ago

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Answer: A cylinder has 3 faces, the circles on the top and bottom and the circular face.

There are two circular edges and two vertices. (The answer, however, depends on your definition of the terms edges, faces, and vertices.)

Detailed explanation

This question does not have a simple answer. A detailed answer and explanation is provided since the topic comes up often. Some of the material in this answer may be beyond what a particular reader is looking for, however, this mathematician feels it is needed to have a full understanding of the answer.

Most of the time when people ask about faces, edges and vertices they are dealing with convex polyhedra where the faces are flat plane regions. The

edges are where two faces meet and the vertices are

where three or more faces meet or where three or more edges meet. The vertices are

points. There is a famous formula known as Euler's formula which says |V| - |E| + |F| = 2.

This simply means the number of vertices minus the number of edges plus the number of faces equals two. For example, a cube has 8 vertices, 12 edges and 6 faces so 8-12+6=2

Now the situation becomes interesting when we want to extend the concepts above to a sphere or a cylinder or to polyhedra in general. In fact we can even look at general topological surfaces since one of the main ideas is that the number of faces, vertices and edges will not change when the surface is deformed. Think of a rubber polyhedron which you can bend or deform any way you want except for cutting. This is one of the main ideas of Topology. (Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects.)

In mathematics it is always important to define our terms before we use them. That is a major reason why this question is so confusing. So if we continue to look at general topological surfaces and define things there, we can always consider convex polyhedra as a special case.

Let's define a face of a topological polygon as a "disc" with a boundary of edges.

We define an edge as a closed curve with two boundary edges-vertices and a vertex is defined simply as a point.

Next we introduce the idea of a connected graph which is a graph (In graph theory, a graph consists of a set of points and a set of lines) where there is a path between any two vertices. We can draw a connected graph on a sphere with no edges crossing one another or on a plane. The graph divides the surface into regions that are known as faces. The graph also creates edges and vertices and the number of edges, vertices and faces obeys Euler's theorem.

Now let's look at a cylinder. We will look at the cylinder as two circular discs at the ends and a rectangle which is folded so that two of the edges are joined to form the side. When learning geometry, some students learn about "nets." The idea is to look at 2 dimensional figures that can be used to create a 3 dimensional figure. For example, if we cut the cube mentioned above into pieces that is an example of a net. Now the net for a cylinder could be thought of as the two discs that make up the end and the bent and joined rectangle that makes up the side. We need a minimum of 3 pieces of tape to join these pieces and form a cylinder. We consider these 3 pieces to be the edges. Furthermore, there are 2 points where the tape pieces meet and these are the 2 vertices. (The piece that joins the rectangle you fold to make the side will meet the top once and the bottom once forming 2 vertices.) You can also think of the 3 pieces of paper as the 3 faces, the top, the bottom and the rectangle. So using Euler's formula (|V| - |E| + |F| = 2) we have |V|=2, |E|=3 and |F|=3 and 2-3+3=2 as expected. This approach is quite nice since it obeys Euler's formula. It is also nice since the surface area of a cylinder is 2(pi* r2) + (2 pi* r)* h which can be easily visualized from this model. This of the rectangle with length 2pi *r and height h and top and bottom with area pi* r2 each.

We could do this with spherical objects as well by cutting them along any great circle.

Then we have 1 vertex where the tape meets to join the halves of the sphere, 1 edge since we need only 1 piece of tape to put the cylinder back together and 2 faces which are two halves created by our cut along a great circle. So this also obeys Euler's formula and we have:

|V| - |E| + |F| = 1 - 1 + 2 = 2

So a sphere can be said to have 1 vertex, 1 edge and 2 faces. Many texts and people would have said a sphere has no edges, no vertices and 1 face. It all depends on how the terms are defined.

If we use the narrow definition that is given at the start of this answer for convex polyhedra, we would say edges are simply straight lines and faces are polygonal regions. This would lead us to say a cylinder has two faces and we are done. From a mathematical point of view, this is not at all useful.

Furthermore, if we go one step farther and say faces are regions and edges are where the regions meet, then a circle has 3 faces, 2 edges and no vertices. Some texts have chosen to adopt this answer and it is important to see where it comes from before accepting or dismissing it.

Some might even consider looking at the circles that form the top and bottom of the cylinders as having an infinite number of edges. This is not unreasonable if we think of the method of approximating the area of a circle using inscribed and circumscribed polygons. We look at regular polygons with more and more sides. In fact if the number of sides is n, we look at the limit as n tends to infinity.

The main point of this very lengthy answer is that in math the definition is the starting point. To answer this question we must define what we mean by edge, vertex, and face.

The answer will certainly change based on the definition. This happens all the time in math.

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