Vertices, Faces and Edges

• 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 a 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. 

What are Edges?

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

• For example, a tetrahedron has 6 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.

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.

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What  are Faces?

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

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

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Here’s a List of 3-D Shapes along with the Number of Faces. 

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Euler’s Formula for 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. 

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

•  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

For example, let us take a cube.

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Let’s list down the number of faces, sides and vertices.

Let’s apply the Euler’s Formula,

=6+8-12

= 14-12  = 2

This is how the Euler’s formula works.

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

Now you Might Think what a Polyhedron is?

A closed solid shape that has flat faces and straight edges is known as a Polyhedron. There are different types of polyhedra. 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 polyhedra generally works for different types of polyhedrons.

Summary

Questions to be Solved

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

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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.

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

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