Vertex

Introduction

Shapes hold an important place in different concepts in mathematics. We know that the boundary or outline of an object is called its shape. We come across many shapes in our daily lives and kids start recognising these shapes even before actually studying about them. The alphabets of English shapes are all shapes of different types. The Sun, the earth and other planets, the mountains and all other things in the world are all of the specific shapes.  There are different parts of shapes that hold significance for their understanding, such as the length and breadth of a 2 – dimensional shape or a length, breadth and height of a 3 – dimensional shape. This means that not every term that is used to discuss and understand shapes might be relevant to all the shapes. However, still, there are some terms, such as vertices and edges that find a place in the discussion of all shapes. Let us learn about the vertex, what it means and how it is relevant in different shapes.

What is a Vertex?

Points of intersection of edges of a polyhedron are known as its vertices. In other words, the meeting point of a pair of sides of a polygon is called its vertex. For example, in the below figure, ABCD, the vertices are A, B, C and D. the plural of the word “ vertex is “ vertices ”. 

There are some other terns as well that are important to recall for further understanding of the vertex. These terms are – 

2 Dimensional shapes – 2 Dimensional shapes or 2D shapes are the shapes that have two dimensions, i.e. length and breadth. For example, square and rectangle are examples of 2 D shapes.

3 Dimensional shapes – 3 Dimensional shapes or 3D shapes are the shapes that have all the three dimensions, i.e. length, breadth and height.  These shapes are called solid shapes. In our everyday lives, we come across many solid shapes. For example, the laptop, our rooms, mobile phones, ice cream, tennis balls etc. are all examples of solid shapes.

Polyhedron – A solid shape bounded by polygons is called a polyhedron.

Faces – Polygons forming a polyhedron are known as its faces. In other words, a face refers to any single flat surface of a 3D shape. For example, observe one side of your room or that of a Rubik’s cube. Every side has two dimensions, be it a length and a breadth, or a breadth and a height or a length and a height. Each side of a room forms its face.

Edges – Line segments common to intersecting faces of a polyhedron are known as its edges. In other words, an edge is a line segment on the boundary joining one vertex (corner point) to another.    It is similar to the sides we have in 2D shapes.

The number of vertices in a shape depends on the type of shape. For instance, the number of vertices in a square and a hexagon are different. Let us now learn about different shapes based on the number of vertices they have.

Shapes with 3 Vertices

3 vertices can be used to form a closed shape. This shape is known as a triangle. So, what is a triangle? A triangle is a polygon that is made up of 3 sides, 3 angles and three vertices. For example, let us have a triangle ABC as shown in the figure below.

In this triangle, 

the three vertices are – A, B and C

the three sides are AB, BC and AC

the three angles are ∠A, ∠B and ∠C

Let us now learn about different shapes that can made using 4 vertices.

Shapes with Four Vertices

4 vertices can be used to make a number of shapes. Such figures are 2 dimensional shapes. Let us see what these shapes are and what are their properties? The shapes that can be made using 4 vertices are known as quadrilaterals. 

What is a quadrilateral?

A quadrilateral is a closed shape that is formed by joining four points among which any three points are non-collinear. In other words, a quadrilateral is a polygon made up of four sides. The order of vertices needs to be kept in mind while naming a quadrilateral.

For instance, below we have a quadrilateral ABCD

This quadrilateral cannot be named as ABDC as the vertices B and D are not adjacent. 

Some of the important quadrilaterals are – 

Square

A square is a quadrilateral that has four equal sides and four right angles. Below is an example of a square ABCD– 

The properties of a square are – 

1. A square has four equal sides, i.e. all the sides of a square are equal. 
2. A square has four right angles
3. A square has two pairs of parallel sides
4. The diagonals of a square bisect each other
5. The diagonals of a square are perpendicular to each other

Rectangle 

A rectangle is a type of quadrilateral that has equal opposite sides and four right angles. Below is a rectangle ABCD – 

The properties of a rectangle are – 

1. A rectangle has two pairs of parallel sides
2. A rectangle has four right angles
3. A rectangle has opposite sides of equal lengths
4. The diagonals of a rectangle bisect each other

Parallelogram

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Below is a parallelogram ABCD – 

The properties of a parallelogram are – 

1. A parallelogram has opposite sides of equal lengths.  
2. The opposite angles of a parallelogram are equal. 
3. A parallelogram has two pairs of parallel sides
4. The diagonals of a parallelogram bisect each other

Trapezium

A trapezium is a quadrilateral in which one pair of opposite sides is parallel. Below is a trapezium ABCD – 

The properties of a trapezium are – 

1. A trapezium has one pair of parallel sides
2. A trapezium has no right angles
3. A trapezium has one pair of opposite sides of equal lengths

Rhombus

A rhombus is a quadrilateral with four equal sides. Below is a rhombus ABCD –

The properties of a rhombus are – 

1. A rhombus has two pairs of parallel sides
2. The opposite angles in a rhombus are equal.
3. A rhombus has four equal sides. This means that all its sides are equal. 
4. The diagonals of a rhombus bisect each other
5. The diagonals of a rhombus are perpendicular each other

Shapes with more than four vertices

3 dimensional shapes are mostly made with more than 4 vertices. Let us learn about some of these shapes.

Cuboid 

A 3D shape having six rectangular faces is called a cuboid. Ex a matchbox, a brick, a book etc. In other words, it is an extension of a rectangle in a 3D plane.

Below we have a general diagram of a cuboid

Faces

A cuboid has 6 rectangular faces, out of which the opposite sides are identical. 

Edges

A cuboid has 12 eddges

Vertx

A cuboid has 8 vertices

Cube

A cuboid whose length, breadth and height are equal is called a cube. Examples of a cube are sugar cubes, cheese cubes and ice cubes. In other words, it is an extension of a square in a 3D plane.

Below we have a general diagram of a cube

Faces

A cube has 6 rectangular faces, out of which all are identical.

Edges

A cube has 12 edges

Vertx

A cube has 8 vertices

Cone

A circular cone has a circular base that is connected by a curved surface to its vertex. A cone is called a right circular cone if the line from its vertex to the centre of the base is perpendicular to the base. An ice-cream cone is an example of a cone

 Below we have a general diagram of a Cone

Faces

A cone has one flat face and one curved surface.  

Edges

A cone has one curved edge.

Vertx

A cone has one vertex.  

 Can there be a shape with no vertices? Let us find out.

Shapes with no vertices

Circles, spheres and cylinders are some of the shapes who do not have vertices.

Circles

Circles are a shape that have no vertices. A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance from a fixed point in the plane. This fixed distance is called the radius of the circle and the fixed point is called the centre of the circle. Below is a circle with centre O.

Sphere

A sphere is a solid formed by all those points in space that are at the same distance from a fixed point called the centre.  In other words, it is an extension of a circle in a 3D plane.

Below we have a general diagram of a Sphere

Faces

A cone has one curved surface.  

Edges

A cone has no edge.

Vertx

A cone has no vertex.  

Cylinder 

A cylinder is a solid with two congruent circles joined by a curved surface. 

Below we have a general diagram of a Cylinder

Faces

A cylinder has one curved surface and two flat faces. 

Edges

A cylinder has two curved edges.

Vertx

A cylinder has no vertices. 

Vertices in Platonic Solids

Platonic Solids are another category of shapes that have different number of vertices depending upon the different shapes of the respective platonic solids. 

A platonic solid is a polyhedron. It is interesting as well as surprising to know that there are exactly five platonic solids. Tetrahedron. These five platonic solids are tetrahedron, cube, octahedron, icosahedron, and dodecahedron. 

Note that in any polyhedron, at least three polygons ( called faces) must meet at the vertex to form a solid angle. Also, the sum of all the plane angles forming the solid angle at a vertex must be less than that of 360^o. Let us start with the simplest regular polygon forming the face of a polyhedron. Clearly, such a regular polygon is an equilateral triangle. Let us now understand about these five platonic solids and their vertices, edges and faces.

Tetrahedron 

Polyhedron or metallic solid whose faces are congruent equilateral Triangles is called the tetrahedron. A unique property of the tetrahedron is that all four vertices are the same distance from each other.

A tetrahedron has – 

4 triangular faces, i.e. F = 4

4 vertices, i.e. V = 4

6 edges, i.e. E = 6

Hexahedron 

Let us now move to the next regular polygon that is square. Six squares form a cube. Cube is the only platonic solid whose every face is a square. Cube is also known as a hexahedron as it has six squares as its faces.

A cube ( hexahedron ) has – 

6 square faces, i.e. F = 6

8 vertices, i.e. V = 8

12 edges, i.e. E = 12

Tetrahedron and cube are platonic solids in which three faces ( regular polygons ) meet at a point to form a vertex.  

Octahedron

Let us now move on to a new platonic solid in which four regular polygons meet at a point to form a vertex. The platonic solid which has four equilateral triangles meeting at each vertex is known as the octahedron.

An octahedron has – 

8 triangular faces, i.e. F = 8

6 vertices, i.e. V = 6

12 edges, i.e. E = 12

Icosahedron

The platonic solid in which five equilateral triangles meet at a point to form a vertex is known as an icosahedron. 

An icosahedron has – 

20 triangular faces, i.e. F = 20

12 vertices, i.e. V = 12

30 edges, i.e. E = 30

Dodecahedron

Now, we have learnt about platonic solids having their faces as equilateral triangles and squares. Let us now discuss about platonic solids whose every face is a pentagon. This platonic solid is known as the Dodecahedron. In a Dodecahedron, three pentagons meet at every vertex.

A dodecahedron has – 

12 triangular faces, i.e. F = 12

20 vertices, i.e. V = 20

30 edges, i.e. E = 30

Relationships between the Vertices, Edges and Faces of Platonic Solids

In order to understand the relationship between the Vertices, Edges and Faces of Platonic Solids. We place the information in the below table.

Platonic Solid	Faces	Edges	Vertices
Tetrahedron	4	6	4
Cube	6	12	8
Octahedron	8	12	6
Icosahedron	12	30	20
Dodecahedron	20	30	12

We have seen above that faces, edges and vertices are important parts of a shape. Can we define a formula to define the relationship between these three components?

Euler’s Formula

Now that we have learnt about the vertices, edges and faces, can we define a relation between them? Let us find out.

Leonhard Euler (1707-1783) was a Swiss mathematician who was one of the greatest and most productive mathematicians of all time. He spent much of his career blind, but still, with the help of scribes, he wrote one paper per week. Euler gave one very popular formula called Euler’s polyhedral formula. So, what was this formula proposed by Euler that defined the relationship between the vertices, edges and the faces of a polyhedron? Let us find out.

According to Euler,

The number of faces (F), the number of vertices (V) and the number of edges (E),  of a simple convex polyhedron are connected by the following formula – 

F + V = E + 2

This is called Euler’s formula.

Euler’s Formula does work only for a polyhedron with certain rules.  The rule is that the shape should not have any holes, and also it must not intersect itself. Also, it cannot be made up of two pieces stuck together, like two cubes stuck together by one vertex.

If all of these rules are properly followed, then this formula will work for all polyhedrons.  Thus this formula will work for most of the common polyhedral.

There are in fact many shapes that produce a different answer to the sum FE.  The answer to the sum FE is sometimes called the Euler Characteristic X.

Let us understand it through an example.

Example

Harry is aware of the fact that a polyhedron has 12 vertices and 30 edges. How can he find 

the number of faces?

Solution  

We will use Euler’s formula to find the polyhedron.

We know that, according to Euler’s formula, the number of faces (F), the number of vertices (V) and the number of edges (E),  of a simple convex polyhedron are connected by the following formula – 

F + V = E + 2

Now, we have been given that Harry is aware of the fact that a polyhedron has 12 vertices and 30 edges

This means that here,

V = 12

E = 30

Putting these values in the Euler’s formula we get.

F + 12 = 30 + 2

⇒ F + 12 = 32

⇒ F = 32 – 12 =20

Hence, the number of faces of the polyhedron = 20

Key Facts and Summary

1. The meeting point of a pair of sides of a polygon is called its vertex.
2. 3 vertices can be used to form a closed shape. This shape is known as a triangle.
3. 4 vertices can be used to make a number of shapes. Such figures are 2 dimensional shapes (2D). Some of the common shapes with 4 vertices are – square, rectangle, parallelogram, rhombus and trapezium.
4. 3 dimensional shapes (3D) are mostly made with more than 4 vertices. Some of the common 3 dimensions shapes are  – cube, cuboid and cone.
5. Circles, spheres and cylinders are some of the shapes who do not have vertices.
6. There are five platonic solids, namely, tetrahedron, cube, octahedron, icosahedron, and dodecahedron. 
7. Euler’s formula states that the number of faces (F), the number of vertices (V) and the number of edges (E),  of a simple convex polyhedron are connected by the following formula 
   F + V = E + 2

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