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There are two broad types of shapes that you study – 2D shapes and 3D shapes. While 2D shapes have length and breadth(width), 3D shapes have three dimensions – length, breadth(width), and height(thickness). As 2D shapes are characterized by their sides and angles, 3D shapes are characterized by faces, edges, and vertices.

Let’s understand what are faces, edges, and vertices and their properties with examples.

## Faces Edges and Vertices

Faces, edges, and vertices are the three properties that define any 3D solid. A vertex is the corner of the shape whereas a face is a flat surface and an edge is a straight line between two faces. The faces, edges, and vertices, differ from each other in appearance and properties.

### What is a Face in a 3D Solid?

A face is an individual surface of a solid(3D) object. It is a polygon on the boundary of a 3D shape. Different 3D objects are characterized by a certain number of faces(flat or curved). For example

- Cubes and cuboids have $6$ flat faces.
- Cones have $1$ flat face and $1$ curved face.
- Cylinders have $2$ flat faces and $1$ curved face.
- Spheres have $1$ curved face.

**Note:** Face is a plane.

### What is an Edge in a 3D Solid?

Edge can be defined as a line where two faces of a 3D shape meet. An edge is a line segment that acts as an interface between two faces or a line segment joining two vertices. For example

- Cubes and cuboids have $12$ edges.
- Cones have $1$ edge.
- Cylinders have $2$ edges.
- Spheres have no edge.

**Note:** Edge is a line.

### What is a Vertex in a 3D Solid?

Vertex(plural vertices) is a point where two or more edges meet. The vertices are the corner points of a 3D shape. For example

- Cubes and cuboids have $8$ vertices.
- Cones have $1$ vertex.
- Cylinders have no vertex.
- Spheres have no edge.

**Note:** Vertex is a point.

## Relation Between Faces, Vertices, and Edges (Euler’s Formula)

The number of faces, edges, and vertices in any shape remains the same irrespective of their size. Also, there is a fixed relation between a number of faces, edges, and vertices in any 3D shape. This relation is called Euler’s formula. Euler’s formula is given by $\text{F} + \text{V} – \text{E} = 2$.

where $\text{F}$, $\text{V}$, and $\text{E}$ are the number of faces, vertices, and edges of a 3D shape respectively.

### Examples on Euler’s Formula

**Example 1:** Find the number of faces of a 3D solid having $6$ vertices and $12$ edges.

$\text{V} = 6$

$\text{E} = 12$

According to Euler’s formula $\text{F} + \text{V} – \text{E} = 2$

$=> \text{F} + 6 – 12 = 2$

$=> \text{F} – 6 = 2$

$=> \text{F} = 2 + 6 = 8$

The number of faces is $8$.

**Example 2:** Find the number of edges of a 3D solid having $20$ faces and $12$ vertices.

$\text{F} = 20$

$\text{V} = 12$

According to Euler’s formula $\text{F} + \text{V} – \text{E} = 2$

$=> 20 + 12 – \text{E} = 2$

$=> 32 – \text{E} = 2$

$=> \text{E} = 32 – 2 = 30$

The number of edges is $30$.

**Example 3:** Find the number of vertices of a 3D solid having $5$ faces and $9$ edges.

$\text{F} = 5$

$\text{E} = 9$

According to Euler’s formula $\text{F} + \text{V} – \text{E} = 2$

$=> 5 + \text{V} – 9 = 2$

$=> \text{V} = 2 + 9 – 5$

$=> \text{V} = 6$

The number of vertices is $6$.

**Example 4:** Is it possible to have a 3D solid having $12$ faces, $18$ edges, and $20$ vertices?

$\text{F} = 12$

$\text{E} = 18$

$\text{V} = 20$

$\text{F} + \text{V} – \text{E} = 12 + 20 – 18 = 14$

Since $\text{F} + \text{V} – \text{E} \ne 2$, therefore 3D solid is not possible with a given number of faces, edges, and vertices.

**Example 5:** Is it possible to have a 3D solid having $12$ faces, $30$ edges, and $20$ vertices?

$\text{F} = 12$

$\text{E} = 20$

$\text{V} = 30$

$\text{F} + \text{V} – \text{E} = 12 + 20 – 30 = 2$

Since $\text{F} + \text{V} – \text{E} = 2$, therefore 3D solid is possible with a given number of faces, edges, and vertices.

## Number of Faces, Edges, Vertices of 3D Shapes

The following table shows the number of faces, edges and vertices of some of the common 3D shapes.

3D Shape | Faces | Edges | Vertices | $\text{F} + \text{V} – \text{E}$ |

Cuboid | $6$ | $12$ | $8$ | $6 + 8 – 12 = 2$ |

Cube | $6$ | $12$ | $8$ | $6 + 8 – 12 = 2$ |

Cone | $2$ | $1$ | $1$ | $2 + 1 – 1 = 2$ |

Cylinder | $3$ | $2$ | $0$ | $3 + 0 – 0= 3$ (Not Applicable) |

Sphere | $1$ | $0$ | $0$ | $1 + 0 – 0 = 1$ (Not Applicable) |

Triangular Prism | $5$ | $9$ | $6$ | $5 + 6 – 9 = 2$ |

Rectangular Prism | $6$ | $12$ | $8$ | $6 + 8 – 12 = 2$ |

Pentagonal Prism | $7$ | $15$ | $10$ | $7 + 10 – 15 = 2$ |

Hexagonal Prism | $8$ | $18$ | $12$ | $8 + 12 – 18 = 2$ |

Octagonal Prism | $10$ | $24$ | $16$ | $10 + 16 – 24 = 2$ |

Triangular Pyramid | $4$ | $6$ | $4$ | $4 + 4 – 6 = 2$ |

Rectangular Pyramid | $5$ | $8$ | $5$ | $5 + 5 – 8 = 2$ |

Pentagonal Pyramid | $6$ | $10$ | $6$ | $6 + 6 – 10 = 2$ |

Hexagonal Pyramid | $7$ | $12$ | $7$ | $7 + 7 – 12 = 2$ |

Octagonal Pyramid | $10$ | $24$ | $16$ | $10 + 16 – 24 = 2$ |

## Practice Problems

- Define the following terms.
- Vertex
- Edge
- Face

- What is Euler’s formula?
- A cube has $6$ faces and $8$ vertices. How many edges it has?
- An icosahedron has $12$ vertices and $30$ edges. How many faces it has?
- An octagonal pyramid has $9$ faces and $9$ vertices. How many edges it has?
- How many faces, edges, and vertices does a sphere have?
- Two cuboids are placed together to form an L-shaped prism. The front and back faces are flat, congruent, 2D shapes.
- Calculate the number of vertices, edges, and faces of the new prism.
- Calculate the value of $\text{V} – \text{E} + \text{F}$, where $\text{V}$, $\text{E}$, and $\text{F}$ are respectively the number of vertices, edges, and faces of the prism.

## FAQs

### What are vertices, faces, and edges?

Vertices are the corners of the 3D shape, where the edges meet. Faces are flat surfaces and edges are the lines where two faces meet.

### What is the relation between the number of faces, edges, and vertices?

Euler’s formula gives the relation between the number of faces, edges, and vertices. According to Euler’s formula $\text{F} + \text{V} – \text{E} = 2$, where $\text{F}$, $\text{V}$, and $\text{E}$ are the number of faces, vertices, and edges of a 3D shape respectively.

### How many edges does a sphere have?

A sphere has zero edges.

### How many faces does a cone have?

A cone has three faces, one flat and one curved.

### Which shape has $5$ faces, $6$ vertices, and $9$ edges?

A triangular prism has $5$ faces, $6$ vertices and $9$ edges.

### What is Euler’s formula?

Euler’s formula gives the relation between the number of faces, edges, and vertices. According to Euler’s formula $\text{F} + \text{V} – \text{E} = 2$, where $\text{F}$, $\text{V}$, and $\text{E}$ are the number of faces, vertices, and edges of a 3D shape respectively.

## Recommended Reading

- 3D Shapes – Definition, Properties & Types
- Nets of 3D Shapes – Meaning, Types & Examples
- What Are 2D Shapes – Names, Definitions & Properties
- Surface Area of a Sphere(Definition, Formulas & Examples)
- Surface Area of a Cylinder(Definition, Formulas & Examples)
- Surface Area of a Cone(Definition, Formulas & Examples)
- Surface Area of A Cube (Definition, Formula & Examples)
- Surface Area of Cuboid (Definition, Formula & Examples)
- Surface Area of a Prism(Definition, Formulas & Examples)
- Surface Area of a Pyramid(Definition, Formula & Examples)
- Volume of a Pyramid(Formula, Derivation & Examples)
- Volume of a Prism(Formula, Derivation & Examples)
- Volume of a Sphere – Formula, Derivation & Examples
- Volume of a Cone(Formula, Derivation & Examples)
- Volume of a Cylinder(Formulas, Derivation & Examples)
- Volume of Cuboid – Formulas, Derivation & Examples
- Volume of a Cube – Derivation, Formula & Examples
- Reference Books
- Sample Papers