Contents

# Face|Definition & Meaning

## Definition

The **flat surfaces** of **geometrical shapes** and **rigid bodies** are called **faces.** For example, a cube has **six faces.**

To understand the **concept** of **faces** we need to understand some **related terms** as well. These terms include **vertices** and **edges.** In the subsequent sections, all three of these terms are **explained** with the help of **suitable examples** and **diagrams.Â **

The following **figure** shows an example of a **cube** which serves as a preliminary introduction to these key concepts of **geometry** when we talk about the **three dimensional** and **two dimensional** shapes.

**Figure 1: Faces, Edges and Vertices of a Cube**

In this **example,** the cube has **eight vertices, twelve edges, and six faces**.Â The calculations become clearer as we introduce these concepts with the help of **examples.**

## What Is a Face?

A **face** in **geometry** is a **three-dimensional** geometric shape’s **planar (flat) surface.** In short, the flat surface of a **solid** body is known as its face. It’s also important to realise that managing two-dimensional forms is physically impossible because everything around us has **three dimensions.**

Consider the **cube** given in the **figure 1**. The **smooth surface** that makes up this cube’s front is known as its **face.** A cube has **six faces**, however, only **three are visible** in the above figure no. 1.

Several faces may be seen on many solid shapes. For instance, a **cuboid has six faces** as shown in the figure below.

**Figure 2: Faces, Edges and Vertices of a Cuboid**

Some geometrical shapes or diagrams are **faceless.** For instance, a pyramid has faces but a **sphere has no face**. So a face is made up of one **flat surface.** In fact, a face is any flat area. The pictures 3 and 4 given below show the **faces of a cone** and **a prism.**

**Figure 3: Faces, Edges and Vertices of a Prism**

The term “face” refers to any of an object’s** distinctive flat surfaces.** There are** four faces on this tetrahedron **as shown in figure 5. Because it could be the face of a polyhedron or the edge of a polygon, the word “side” is not very precise. A solid’s face is any one flat surface. Solids can have **several faces.** A face is a flat surface (a planar region) that forms part of a **solid object’s border**.

## What Is a Vertex?

In any **three-dimensional** or two dimensional shape, **vertices** are the **locations** where **two or more line segments or edges cross**. They can be thought of like a **corner.** A single such corner is called a vertex and multiple such corners are termed as vertices.

For instance, the **cube** shown in the figure no. 1 has **eight vertices**. One of the vertices is hidden in this figure and only seven are displayed. The** cone shape** shown in the figure below, however, only has **one vertex**.

**Figure 4: Faces, Edges and Vertices of a Cone**

In short, **vertex** is another name for a **corner.** Many solids have many vertices or a** large number of vertices**.

## What Are Edges?

The **line segments known as edges** act as the **meeting places for the faces** of a shape and** join one vertex to the next**. They may be used to describe both two dimensional and three dimensional **geometrical shapes.** Some shapes, like **hemispheres,** have **curved edges** even though many forms have straight edges and lines.

A **cube,** shown in the figure 1, hasÂ **twelve straight line edges**. Nine out of these twelve are visible in the diagram while three are hidden. A quick glance reveals that this **cube’s faces intersect one another in a straight line**.

So we can say that **a segment of a line joining two faces** or **connecting two vertices** (corner points) on a polygon’s boundary is referred to as an **edge.**Â Many solid forms have several edges. As another example, the **tetrahedron** shown in the following figure has **six edges.**

**Figure 5: Faces, Edges and Vertices of a Tetrahedron**

## Euler’s Theorem

The **Euler theorem,** named after Leonhard Euler, is one of the most important **mathematical theorems****.** The link between a polyhedron’s face, vertex, and edge counts is established by the theorem. To understand Euler’s theorem, we first need to understand the term **Polyhedron.** Euler’s polyhedron formula is accurate for the majority of polyhedron types.

### What Is a Polyhedron?

A **closed-space** object formed **entirely** of **polygons** is called a **polyhedron.** The English term “polyhedron” is derived from the Greek words **“poly”** and **“hedron,”** which combined denote a **base** or **seat.**

In other words, a **closed solid shape** with flat sides and straight edges is known as a **polyhedron.** There are several varieties of polyhedra. Due of its rounded edges, a **cylinder** is **not** a **polyhedron;** in contrast, a cube is an illustration of one.

Polyhedrons have faces, edges, and vertices. The **edges** of a **polyhedron** are made up of **line segments** that span two faces. **Vertices** are the **intersections** of three or more edges. In summary, aÂ polyhedron is a three-dimensional solid with faces acting as its only **boundary.**

### Mathematical Form of Euler’s Theorem

**Euler’s Theorem** shows how a polyhedron’s **faces, vertices, and edges are connected**. According to this theorem, for many solid shapes, the** total of the number of faces plus the number of vertices minus the number of edges is always equal to two**.

Mathematically:

**( Faces ) + ( Vertices ) – ( Edges ) = 2**

Let’s test this condition for our initialÂ **example**Â of theÂ **cube.**Â We already know that a cube has**Â twelve edges, eight vertices, and six faces**. So putting these values in the above formula:

**( 6 ) + ( 8 ) – ( 12 ) = 14 – 12 = 2**

Hence, the Euler’s theorem holds true for the case of cube. Its important to reiterate that Euler’s Formula only holds true for the Polyhedrons.

## Numerical Example of Faces

Given the following **geometrical shape (Octahedron)**, find the **number of faces, edges and vertices**. Also check if the **Euler’s theorem holds true** for this shape or not?

Figure 6: Numerical Example for Faces, Edges and Vertices (Octahedron)

### Solution

The given figure has** eight faces, twelve edges and six vertices.**

Putting these values in the **Euler’s formula:**

**( 8 ) + ( 6 ) – ( 12 ) = 14 – 12 = 2**

Hence, the **Euler’s theorem holds true** for the case of cube.

*All figures and charts have been constructed using GeoGebra.*