 # 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.

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.

## 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.

## 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.

## 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,

## 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:

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:

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.

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

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.