Faces, Edges And Vertices of 3D Shapes
Last Updated :
23 Jul, 2025
Faces, Edges, And Vertices of 3D Shapes: Faces, Edges, and Vertices are the three key components that are used to define various 3D objects. They have different dimensions like length, width, and height.
In geometry, Faces are the flat surfaces of a 3D shape. They are bounded by edges and are what give the shape its appearance. Edges are the straight lines where two faces of a 3D shape meet. They form the boundaries between faces and help define the shape's overall structure. Vertices (singular: vertex) are the points where the edges of a 3D shape meet. They are essentially the corners of the shape.
In this article, we are going to learn about the faces, edges, and vertices of different 3D shapes in detail.
What are Faces?
A face of a shape is defined as the flat surface of a solid 3D Shape. It can be referred to as the outer surface of a solid object, whether it be a curved face or a straight face.
The figure given below is a cuboid that has six faces, i.e., ABCD, EFGH, ADHE, AEFB, BFGC, and DHGC.
What are Edges?
An edge is defined as the line segment where the faces of a solid meet. Even though many shapes have straight edges and lines, some shapes, like hemispheres, have curved edges.
What are Vertices?
In geometry, a vertices are point of intersection of two or more curves, lines, or edges. They are represented as points and denoted by letters such as A, B, C, D, etc.
The figure given below is a cuboid that has eight vertices, i.e., A, B, C, D, E, F, G, and H.
What are Polyhedrons?
Polyhedron is referred to as a three-dimensional solid that is made up of polygons. It consists of flat polygonal faces, straight edges, and sharp corners called vertices.
Some examples of polyhedrons are cubes, pyramids, prisms, etc.
Types of Polyhedrons
Different types of polyhedrons have been discussed in the table below:
| Types of Polyhedron | Description | Examples |
|---|
| Regular Polyhedron | A polyhedron whose faces are regular, congruent polygons. | Tetrahedron, Cube, Regular Octahedron, Regular Dodecahedron, Regular Icosahedron |
| Irregular Polyhedron | A polyhedron made up of irregular polygonal faces that are not congruent to each other. | Pentagonal Pyramid, Triangular Prism, Octagonal Prism |
| Convex Polyhedron | A polyhedron where the line segment joining any two points lies within its interior or surface, and the surface does not intersect itself. | Cube, Regular Tetrahedron, Prism |
| Concave Polyhedron | A non-convex polyhedron where at least one line segment joining two points of the polyhedron lies outside its surface. | Star-shaped Polyhedrons, Concave Cubes |
Note : As cones, cylinders, and spheres do not have polygonal faces and curved surfaces, they are non-polyhedrons.
Faces, Edges And Vertices of 3D Shapes
Now we will find the vertices, faces and edges of various solid shapes with the help of a table given below.
Shape | Figure | Vertices | Edges | Faces |
|---|
Cube |  | 8 | 12 | 6 |
Cuboid |  | 8 | 12 | 6 |
Cone |  | 1 | 1 | 2 |
Cylinder |  | 0 | 2 | 3 |
Sphere |  | 0 | 0 | 1 |
Triangular Prism |  | 6 | 9 | 5 |
Rectangular Prism |  | 8 | 12 | 6 |
Square Prism |  | 8 | 12 | 6 |
Pentagonal Prism |  | 10 | 15 | 7 |
Hexagonal Prism |  | 12 | 18 | 8 |
Triangular Pyramid |  | 4 | 6 | 4 |
Rectangular Pyramid |  | 5 | 8 | 5 |
Square Pyramid |  | 5 | 8 | 5 |
Pentagonal Pyramid |  | 6 | 10 | 6 |
Hexagonal Pyramid |  | 7 | 12 | 7 |
Relation Between Faces, Edges And Vertices of 3D Shapes
The relationship between vertices, faces, and edges can be determined using Euler's formula. Euler's formula states that for any convex polyhedron, the sum of the number of faces (F) and vertices (V) is exactly two greater than the number of edges (E).
F + V = 2 + E
where,
F is the number of faces,
V is the number of vertices,
E is the number of edges
Euler's formula is applicable for closed solids that have flat sides and straight edges, such as cuboids. It is not applied to solids that have curved edges, such as cylinders, spheres, etc.
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Faces, Edges And Vertices of 3D Shapes Examples
Example 1: Verify Euler’s formula for the heptahedron.
Solution:
A heptahedron has 7 faces, 10 vertices, and 15 edges.
From Euler’s formula, we have,
F + V = 2 + E
⇒ 7 + 10 = 2 + 15
⇒ 17 = 17
Hence, Euler’s formula is verified for the heptahedron.
Example 2: Calculate the number of edges of a polyhedron if it has 6 vertices and 9 faces.
Solution:
Given data,
V = 6 and F = 5
From Euler’s formula, we have,
F + V = 2 + E
⇒ 5 + 6 = 2 + E
⇒ 11 = 2 + E
⇒ E = 11 – 2 = 9
Hence, the given polyhedron has 9 edges.
Example 3: Calculate the number of vertices of a polyhedron if it has 5 faces and 8 edges.
Solution:
Given data,
F = 5 and E = 8
From Euler’s formula, we have,
F + V = 2 + E
⇒ 5 + V = 2 + 8
⇒ 5 + V = 10
⇒ V = 10 – 5 = 5
Hence, the given polyhedron has 5 vertices.
Conclusion
Understanding the faces, edges, and vertices of 3D shapes is key to understand the basics of geometrical figures. These elements help define the structure and properties of three-dimensional objects, from cubes and pyramids to spheres and cylinders. Knowing how to count faces, edges, and vertices allows us to better visualize and work with shapes in both math and real-life situations.
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