Objects have diverse physical characteristics that distinguish them: length, width, and diameter being a few of them. However, regardless of how different their dimensions are, they all take up space and have three dimensions. As a result, they are referred to as solids or three-dimensional shapes.
Figures with two dimensions – length and breadth – can be depicted on a plane (for instance, a piece of paper). They are often referred to as two – dimensional figures or planar shapes.
The three qualities that define any three-dimensional solid are vertices, faces, and edges.
Knowledge of these entities helps us define a solid accurately.
This article will explore the definitions of faces, edges, and vertices. Along with that, we would also cover Euler’s formula and its characteristics. In the end, there are some solved examples and questions to polish and embed your mathematical concepts.
|Table of contents|
|What are Vertices?|
|What are Edges?|
|What are Faces?|
|Questions to Polish your Concepts|
What are Vertices?
Vertex ( plural vertices ) refers to the topmost point of a geometric figure. Since it is the highest point of a shape or a solid, it is also the farthest away from the figure’s base.
Another definition of a vertex is the point of meeting two sides of a plane figure. It can also be defined as a point of a geometrical figure that is the same for three or more sides. Every figure or shape made up of line segments has a set of vertices within it. They are the starting and ending points of edges too.
Here is a list of various polyhedrons with their corresponding number of vertices:
|Polyhedron’s name||Number of Vertices|
|Rectangular Prism||8 vertices|
|Triangular Prism||6 vertices|
|Hexagonal Prism||12 vertices|
|Pentagonal Prism||10 vertices|
|Square pyramid||5 vertices|
|Octagonal prism||16 vertices|
|Triangular pyramid||4 vertices|
|Rectangular pyramid||5 vertices|
|Hexagonal pyramid||6 vertices|
|Octagonal Pyramid||7 vertices|
What are Edges?
Edges of a figure are referred to as the outline of a shape or a solid. They can also be defined as the set of lines that join together to form vertices of various figures.
When we talk about the relation between edges and vertices of a figure, we must acknowledge that there exist considerable contrasts between two-dimensional and three-dimensional figures.
For instance, in 2-D figures like squares and triangles – there are four edges and four vertices; and three edges and three vertices, respectively. However, in the case of a 3-D solid like a cube – there are 12 edges and 8 vertices. Hence, even though the edges and vertices can be identical in number in the case of 2-D shapes, it is not the case when we talk about 3-D figures.
Here is a list of various polyhedrons with their corresponding number of edges:
|Polyhedron’s name||Number of edges|
|Rectangular Prism||12 edges|
|Triangular Prism||9 edges|
|Hexagonal Prism||18 edges|
|Pentagonal Prism||15 edges|
|Square pyramid||8 edges|
|Octagonal prism||24 edges|
|Triangular pyramid||6 edges|
|Rectangular pyramid||8 edges|
|Hexagonal pyramid||12 edges|
|Octagonal Pyramid||16 edges|
What are faces?
One of the essential components of a geometrical figure, the face, is defined as any specific flat surface of a solid object.
The face of a 3-D figure is usually a 2-D shape. For instance, the faces of a cube would be in the shape of squares. The exception to this rule is a sphere with only one face with no vertices or edges.
The faces of 3-D figures can be of various kinds – ranging from curved faces to flat faces. We often observe different 3-D figures that comprise a combination of these kinds of surfaces. For example, a cone (3-D figure) has one flat face and one curved face.
Here is a list of polyhedrons with their corresponding number of faces:
|Polyhedron’s name||Number of Faces|
|Cone||2 faces (1 flat face and 1 curved face)|
|Sphere||1 continuous face|
|Cylinder||3 faces (2 flat faces and 1 curved face)|
|Rectangular prism||6 faces|
|Triangular prism||5 faces|
|Hexagonal prism||8 faces|
|Pentagonal prism||7 faces|
|Square pyramid||5 faces|
|Octagonal prism||10 faces|
|Triangular pyramid||4 faces|
|Rectangular pyramid||5 faces|
|Hexagonal pyramid||7 faces|
|Octagonal pyramid||9 faces|
Now that we are clear with what vertices, edges, and faces are – we can now move on to how they are related. Using Euler’s formula, we calculate the relationship between various geometrical figures’ vertices, faces, and edges.
It’s important to remember that the formula only applies to closed solids with flat sides and straight edges and is irrelevant for figures with curved edges. For instance, we can undoubtedly use Euler’s formula for cubes and cuboids, but it is not used in the case of cylinders.
The Euler theorem, named after Leonhard Euler, is one of the most important mathematical theorems. This theorem helps us calculate and analyze various components of a 3-D geometrical figure. Using this formula, we need not count the edges, vertices, and faces of a figure every time we need the numbers.
According to Euler’s theorem,
F+V-E = 2, where F represents the number of faces, V represents the number of vertices, and E represents the number of edges.
Let us now take a real-life example and apply Euler’s formula ourselves.
We will apply Euler’s formula to a tetrahedron, a 3-D geometrical figure with no curved surface.
|Entities of tetrahedron||Tetrahedron|
|Number of Faces||4|
|Number of Vertices||4|
|Number of Edges||6|
Now according to the Euler’s formula, we have:
F+V-E = 2 (where F = faces, V= vertices, and E= edges)
8-6 = 2
2 = 2
Since LHS=RHS, we can say that a tetrahedron follows Euler’s formula. Hence, this theorem is used to find out any one of the three missing entities of the figure.
F+V-E = sigma, which is Euler’s characteristic of being any integer value
But specifically for convex polygons, the value is considered as 2.
- Find the number of faces in a polyhedron that has 8 vertices and 12 edges. What is the shape you obtain?
- According to the Euler’s formula, we get
F+V-E = 2
Rewriting this, we get:
The polyhedron thus obtained with 6 faces is a cube.
- Prove the Euler’s formula for a prism?
- According to the Formula, F+V-E = 2.
Faces in a prism = 5
Edges in a prism = 9
Vertices in a prism = 6
Substituting the value in the formula, we get:
11-9 = 2
- Can a polyhedron have 20 faces, 10 edges, and 15 vertices?
- If a polyhedron has 20 faces, 10 edges, and 15 vertices – it should be able to satisfy Euler’s formula.
Here, the number of faces F = 20
Number of edges E = 10
Number of vertices V = 15
Hence, by Euler’s formula:
LHS: 20+15 = 35
RHS: 10+2 = 12,
which is not true because 35≠12
Hence, Euler’s formula is not satisfied. This proves that such a polyhedron does not exist.
Questions to Polish your Concepts
- Prove Euler’s formula for octahedra.
- Using Euler’s formula find the unknown:
3. Can a polyhedron have for its faces:
(i) 2 triangles?
(ii) 5 triangles?
(iii) a square and three triangles?
4. Is a square prism and a cube the same thing?
We come across objects of various sizes and shapes in our day-to-day lives, and these objects are nothing but polyhedrons with fixed dimensions and properties. Properties such as Faces, Edges, and vertices. A vertex is the intersection of two or more line segments.
A face of a figure is usually defined as any individual flat surface of a solid object, and a point where two faces meet can be characterized as an edge. All three entities are linked to each other, and the relation between them is given by F+V-E=2, which is Euler’s formula.
1. Can a polyhedron have 10 faces, 20 edges, and 15 vertices? Why or why not?
For a 3-D figure to have a certain number of faces, edges, and vertices – Euler’s formula needs to be satisfied. In this case, too, we have to check whether the formula is obeyed to confirm if a polyhedron with these characteristics can exist or not.
Since 10+15-20 is not equal to 2, Euler’s formula is not satisfied. Hence, a polyhedron cannot exist with 10 faces, 20 edges, and 15 vertices.
2. Are vertices corners of a figure?
Yes, vertices can also be referred to as the corners of a figure. Since they are the point that is farthest from the base and is responsible for connecting the edges of the figure, vertices adhere to the definition of a corner.
3. How many edges, faces, and vertices does a cylinder have?
The cylinder comprises of the following entities:
- Faces: a cylinder comprises two faces – a curved face ( the cylindrical body ) and a flat face ( the base and top of the cylindrical body ).
- Vertices: a cylinder has no vertices.
- Edges: a cylinder has no edges.
4. What is the difference between vertices and an angle?
Though they might seem the same in definition, vertices and angles are quite different in nature. While angles are a meeting point for two rays, the vertices are the common meeting point for two line segments. Apart from that, angles measure the inclination of the rays that meet at a certain point. Vertices calculate no such numbers and are simply the point of intersection for two crossing line segments.
5. What are polyhedrons? What do we call these figures with identical faces?
The 3-D shapes with 2-D polygons in the shape of their faces are referred to as polyhedrons. Geometric figures with identical faces are called regular polyhedrons. All the faces of these polyhedrons are completely identical in looks, and there exist no irregularities whatsoever.
6. Does Euler’s formula hold true for all kinds of shapes?
No, Euler’s formula does not hold true for all kinds of shapes and figures. Apart from the mandatory absence of all kinds of curved surfaces, figures should be following the below-mentioned rules to be fit Euler’s formula:
- The figure should not consist of any sort of hole within itself.
- The figure should not intersect itself at any portion.
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