Contents

# Vertices|Definition & Meaning

## Definition

Vertices are the plural of **vertex**, which is any one of the **corner** points of a shape. Therefore, a vertex is a point where two or more **line** segments **join** together, so it follows that different **shapes** have different numbers of vertices (e.g., a **triangle** has **3** vertices while a **square** has **4**, a **pyramid** has 5 while a **cube** has **8**, etc.). They are one of the defining **features** of shapes.

**Figure 1** shows the **vertices** of a triangle and a square.

## Polygon

A polygon is a **two-dimensional** shape bounded by straight lines. It is a **plane** or flat shape, not having curved surfaces. A polygon consists of **sides,** also known as **edges**. The points or **corners** where two edges meet are called the **vertices**.

Different **types** of **polygons** include a triangle, **quadrilateral**(square, rectangle, parallelogram), pentagon, **hexagon**, heptagon, octagon, nonagon, and **decagon**.

## Polyhedron

A polyhedron is a **three-dimensional** solid with straight **edges**, sharp **vertices**, and flat polygonal **faces**. Spheres and cones are not considered polyhedrons as they do not have polygonal faces. **Cubes** and **pyramids** are examples of polyhedrons.

## Other Defining Features of Polygons and Polyhedrons

The number of **vertices** in a polygon or a polyhedron defines its **structure**. Other important **features** that define their shape’s symmetry are their **edges**, faces, and **sides**. These are discussed as follows.

### Faces

The term “**faces**” is only used for **polyhedrons**. The outer flat **surfaces** of a polyhedron are its faces. A **cube** has six faces, and a **pyramid** has five faces. An individual **face** of a polyhedron forms its respective **polygon**.

**Figure 2** shows the faces of a **rectangular prism** that has **eight** vertices.

A **rectangular** prism has **six** faces with **four** lateral faces and **two** faces known as rectangular bases. A **base** shape of a prism defines the **type** of prism.

### Edges

An edge of a polygon is the **line segment** joining any two adjacent **vertices**. For example, a **hexagon** consists of **six** edges. The edges of a **polyhedron** are simply the line segments around which any number of **faces** (at least two) **meet**.

**Figure 3** shows the five edges of a **pentagon**(polygon) and some of the **edges** of a **tetrahedron**(polyhedron). A tetrahedron has a total of **eight** edges.

### Sides

The term “**side**” means different for a polygon and a **polyhedron**. The **edge** of a **polygon** can be referred to as the **side** of a polygon. A polyhedron’s **face** is also called one of its sides.

## Adjacent Vertices

Adjacent vertices are the vertices **next to** each other. Two vertices are **adjacent** if they are joined by a **side** or an **edge** in a polygon. A **hexagon** is shown in** figure 4**.

The vertices **P** and **Q**, **R** and **S**, and **T** and **U** are **adjacent** to each other as they are connected by an edge. The vertices **P** and **R** are **non-adjacent,** as there is no common **edge** between them.

## Opposite Vertices

Opposite **vertices** are defined as the two vertices **opposite** to each other.

If two **opposite vertices** of a polygon are joined by a **line segment**, then that line segment is known as the **diagonal**. A triangle has no diagonal, as there are no opposite vertices in a **triangle**.

**Figure 5** shows a **rectangle** LMNO.

The two vertices, **L** and **N**, and **M** and **O** of the rectangle are **opposite** to each other. A **diagonal** can be formed by **joining** any of these two opposite vertices.

### Diagonal Formula

We can evaluate the **number** of **diagonals** of a polygon by using the **formula** given as:

Number of Diagonals = [m(m-3)] / 2

Where “**m**” is the number of **vertices** of an **m-sided** polygon.

For example, to calculate the number of **diagonals** of a **nonagon**, the number of **vertices** or the sides of the nonagon should be known. A nonagon has nine sides and **nine** vertices, so:

m = 9

Putting the value of **m** in the above **formula** gives the total number of diagonals of a nonagon as:

Number of Diagonals of a Nonagon = [9(9-3)] / 2

Number of Diagonals = 9(6) / 2 = 54 / 2

Number of Diagonals = 27

Hence, a **nonagon** has a total of **27** diagonals.

## Euler’s Formula

**Euler’s** formula gives a **relation** between the edges, vertices, and **faces** of a polyhedron. It states that the **sum** of the number of **vertices** and the number of **faces** of a polyhedron is equal to the number of **edges** plus **two**. It is given as follows:

V + F = 2 + E

Where **V**,** F**, and** E** are, respectively: the number of **vertices**, the number of **faces,** and the number of **edges**. This **formula** can also be written as:

V + F – E = 2

Hence, the number of edges **subtracted** from the **sum** of the number of vertices and faces of a **polyhedron** always equals **two**.

For example, a **triangular prism** has **4** faces, **4** vertices, and **6** edges. Putting the values in **Euler’s** formula gives:

4 + 4 = 2 + 6

8 = 8

A **triangular** prism is also known as a **tetrahedron**.

Euler’s formula works for almost all **polyhedrons** except the solids with **holes** in them.

## Platonic Solids

A platonic solid is a solid having all the **faces** as the same **regular** polygons. A regular **polygon** has all the sides of equal **length** and all its angles **congruent** to each other. Also, an **equal** number of polygons are joined at each **vertex** of a **platonic** solid.

The **five** major platonic solids are the **tetrahedron**, cube, **octahedron**, dodecahedron, and **icosahedron**.

A **tetrahedron** has four vertices, with each **vertex** joining three **triangles**. It has four faces and six edges. A cube has **eight** vertices joining three **squares** at each vertex. It has **six** faces and **twelve** edges.

An **octahedron** has six **vertices** with four **triangles** joined at each vertex. It has eight faces and twelve edges. An **icosahedron** has five triangles meeting at each of its **twelve** vertices. It consists of twenty **faces** and thirty edges.

## Vertex of an Angle

An **angle** is formed by joining two rays at a **common** point. This common point is also known as the **vertex** of the angle. **Figure 6** shows the demonstration of the **vertex** of an **acute** angle.

## Examples of Vertices

### Example 1 – Identifying Vertices

Identify the number of **vertices** of a heptagon. Calculate the number of **diagonals** that can be formed by the **opposite** vertices of a heptagon and **draw** these diagonals. Also, identify any two pairs of **adjacent** vertices and opposite vertices of the **heptagon**.

### Solution

A **heptagon** has seven **edges** or sides. It has a total of **seven vertices**. The number of **diagonals** of a heptagon can be calculated by using the **formula**:

Number of Diagonals = [m(m-3)] / 2

Here, m = 7, putting the value of m gives:

Number of Diagonals = [7(7-3)] / 2

Number of Diagonals = [7(4)] / 2 = 28 / 2

**Number of Diagonals = 14**

Hence, a **heptagon** has a total of **14 diagonals** shown in **figure 7**.

The vertices **A** and **B** are **adjacent** vertices, and the vertices **A** and **D** are two **opposite** vertices of the heptagon.

### Example 2 – Using Euler’s Formula To Calculate the Number of Vertices of a Platonic Polyhedron

Calculate the number of **vertices** of a dodecahedron having twelve **faces** and thirty **edges**.

### Solution

**Euler’s** formula is given as follows:

V + F – E = 2

For a **dodecahedron**:

F = 12 , E = 30

For the number of **vertices V**, the formula becomes:

V = 2 – F + E

Putting the values of **F** and **E** gives:

V = 2 – 12 + 30

V = 2 + 18

**V = 20**

Hence, a dodecahedron has **twenty vertices**.

*All the images are created using GeoGebra.*