Vertices, Faces and Edges

What is an Edge, Vertex and a Face?

  • A vertex in a geometrical figure can be defined as a corner.

  • A line segment between faces is known as an edge.

  • A single flat surface is known as face.

[Image will be Uploaded Soon]            

What are Vertices?

  • A point where two or more line segments meet is known as a vertex.

  • The plural of vertex is vertices.

  • In simpler words, we can say that a vertex is a corner.

  • For example, a tetrahedron has 4 vertices and a pentagon has 5 vertices.

[Image will be Uploaded Soon]            

Here’s a List of Shapes along with the Number of Vertices. 

3D Shape Vertices

Number of Vertices (V)

Cube

8 vertices

Cone

1 vertex

Sphere

0 vertex

Cylinder

0 vertex

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

Pentagonal pyramid

4 vertices

Hexagonal pyramid 

7 vertices

Octagonal pyramid

9 vertices


What are Edges?

  • An edge in a shape can be defined as a point where two faces meet.

  • For example, a tetrahedron has 4 edges and a pentagon has 5 edges.

  • The line segments that form the skeleton of the 3D shapes are known as edges.

  • For a polygon, we can say that an edge is a line segment on the boundary joining one vertex (corner point) to another.

  •  A Tetrahedron Has 6 Edges

  • For polyhedron shapes a line segment where two faces meet is known as an edge.

[Image will be Uploaded Soon]

Here’s a List of Shapes along with the Number of Edges.

Shape

Number of Edges(E)

Cube

12 edges

Cone

1 edges

Sphere

0 edge

Cylinder

3 edges

Rectangular prism

12 edges

Triangular prism

9 edges

Hexagonal prism

18 edges

Pentagonal prism

12 edges

Square pyramid

8 edges

Octagonal prism

24 edges

Triangular pyramid 

6 edges

Rectangular pyramid

8 edges

Pentagonal pyramid

10 edges

Hexagonal pyramid 

12 edges

Octagonal pyramid

16 edges


What do you Mean by Faces?

  • A face of a figure can be defined as the individual flat surfaces of a solid object.

  • Example, a tetrahedron has 4 faces one of which is not visible.

[Image will be Uploaded Soon]                                 

Here’s a List of Shapes along with the Number of Faces. Faces of 3d Shapes are Given Below:  

Shape

Number of Faces

(Faces of 3d shapes)

Cube

6 faces

Cone

2 faces

Sphere

1 face

Cylinder

3 faces

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

Pentagonal pyramid

4 faces

Hexagonal pyramid 

7 faces

Octagonal pyramid

9 faces


Euler’s Formula for Polyhedron:

What is Euler’s Formula for Types of Polyhedron?

  • The Euler theorem is known to be one of the most important mathematical theorems named after Leonhard Euler.

  •  The theorem states a relation of the number of faces, vertices, and edges of any polyhedron. 

  • The Euler’s formula can be written as F + V = E + 2, where F is the equal to the number of faces, V is equal to the number of vertices, and E is equal to the number of edges.

  •  The Euler’s formula states that for many solid shapes the number of faces plus the number of vertices minus the number of vertices is equal to 2.

Euler’s Formula:

F + V − E = 2


For example ,

Let us take a cube,

[Image will be Uploaded Soon]

Let’s List Down the Number of Faces, Sides and Vertices.

3d Shapes Faces Edges Vertices

CUBE

No of faces

6

No of Edges

12

No of Vertices

8


Let’s apply the Euler’s Formula,

Euler’s Formula:

F + V − E = 2

=6+8-12

= 14-12  = 2

This is how the Euler’s formula works.

Note: The Euler's formula for polyhedron generally deals with shapes called Polyhedron shapes. 

Now You Might Think What is a Polyhedron?

Here’s what is a polyhedron,

A closed solid shape which has flat faces and straight edges is known as a Polyhedron. There are different types of polyhedron. A cube can be an example of a polyhedron whereas as a cylinder has curved edges it is not a polyhedron. Euler’s formula for polyhedron generally works for types of polyhedrons.

Summary:

Name

How to Remember?

Vertex

Corner

Edge 

Straight Line

Face

Surface

Questions to be Solved:

Question 1) Find the number of faces, edges of 3d shapes and vertices in the figure given below:

[Image will be Uploaded Soon]

Solution) The figure given above is a square pyramid.

As we can see from the figure, a square pyramid has 5 faces, 5 vertices and 8 edges.

Question 2) Find the number of faces, edges and vertices in the figure given below:

[Image will be Uploaded Soon]

Solution) The figure given above is a cylinder.  And as we know that a cylinder has 2 faces, 0 vertices and 0 edges.

Question 3) Show how the Euler’s formula works for a cube.

[Image will be Uploaded Soon]

Solution) 

Let’s List Down the Number of Faces, Sides and Vertices of Polyhedron Shapes.

3-D Solid

CUBE

No of faces

6

No of Edges (edges of 3d shape)

12

No of Vertices(3d shapes vertices)

8


Let’s apply the Euler’s Formula,

Euler’s Formula:

F + V − E = 2

=6+8-12

= 14-12

= 2


FAQ (Frequently Asked Questions)

Q1. What is the Relation Between Faces Vertices and Edges and How Many Faces, Edges and Vertices do 3d Shapes have?

Ans. The edges of any figure can be defined as are edges where the faces meet each other. The vertices can be defined as the corners of the figure. From Euler's Formula we know that if we add the number of faces and vertices of the figure together and then subtract the number of edges, the answer we will get will be equal to 2. 

The formula can be written as 

F + V - E = 2

Here are the 3d shapes faces edges vertices ,

Cylinders and prisms have two bases that are both parallel and congruent. An edge is a line segment where two faces meet.

Q2. What are Edges and Vertices and What Shape has 5 Faces 9 Edges 6 Vertices?

Ans. We can Define a Face as a Flat Surface.

An edge in a shape can be defined as a point where two faces meet 

For example, a tetrahedron has 4 edges and a pentagon has 5 edges,the line segments that form the skeleton of the 3D shapes are known as edges.

For a polygon, we can say that an edge is a line segment on the boundary joining one vertex (corner point) to another. And vertex is a corner where edges meet and the plural of vertex is vertices. A Triangular prism is the shape that has 5 Faces, 6 Vertices and 9 Edges.