 In mathematics, 3-Dimensional shapes are a very important topic of geometry. In our day to day life, we deal with lots of 3-D objects which have length, breadth, and depth. All objects are of different sizes and shapes. For example, balls, ice-cream cones, books, etc.

They all occupy space and are 3-D. Sometimes they are referred to as solids too. In this article, we will discuss various aspects of 3-D shapes with definitions and examples.

They are made up of vertices, faces, and edges. We can say that vertices, faces and edges are the three main properties that define any 3-dimensional shapes of geometry. Let’s see what does each term mean in very simple language. The pointy bits or the corners of a shape where edges meet are known as vertices.

Lines around the shape are known as edges. The flat sides of a shape that you touch when you hold a shape are known as faces. For example, the cube has 6 square faces, 8 vertices, and 12 edges, while the sphere has 0 faces, 0 edges, and 0 vertices.

### Vertices

The vertex is written as vertices in plural form, usually denoted by capital letters such as E, P, Q, S, Z, etc. Vertex is a point where two or more curves, lines or edges of a shape meet. More specifically, a point where two or more lines of a polygon meet to form an angle or the corner is called vertex.

### Faces

A flat surface or a plane region that creates part of the boundary of a solid object is known as a face. It is also known as the side of an object. For example, a cube has 6 faces, and a cylinder has 3 faces.

### Edges

In geometry, an edge is a line that joins two vertices of a polygon. It is also known as a side. In simple language, an edge is a line segment on the boundary, or an edge is a line segment where two faces meet. For example, a pyramid has 8 edges, and cuboids have 12 edges.

Knowing the faces, vertices, edges properties of any objects lays the foundation for various industries such as architecture, interior design, engineering and more.

### Consolidated table of vertices, faces and edges of 3-Dimensional shapes

Let’s see vertices, faces and edges of all known 3-dimensional shapes and see how they differ from one another.

 Object Name Vertices Faces Edges Cube 8 6 12 Cuboids 8 6 12 Cylinder 0 3 2 Sphere 0 1 0 Cone 1 2 1 Hemisphere 0 2 1 Tetrahedron 4 4 6 Rectangular Prism 8 6 12 Triangular Prism 6 5 9 Hexagonal Prism 12 8 18 Pentagonal Prism 10 7 12 Square Pyramid 5 5 8 Octagonal Prism 16 10 24 Triangular Pyramid 4 4 6 Rectangular Pyramid 5 5 8 Pentagonal Pyramid 4 4 10 Hexagonal Pyramid 7 7 12 Octagonal Pyramid 9 9 16

### Euler’s Formula

Euler’s formula can be used to find out the relation between vertices, faces, and edges. Euler’s formula has been named after a famous mathematician Leonhard Euler. He invented this formula, and hence formula has been given his name.  The formula can be used only for closed solids with flat faces.

In short, for this formula to work, the shape must not have any holes, and it must not intersect itself. For example, cube, cuboids, etc. it can’t be applied for sphere, cylinder or cone. It can not be made up of two pieces stuck together, such as two cubes stuck together by one vertex.

It usually works for most of the common polyhedral which we have heard of.

Euler’s formula is,

F + V –E = 2

Where,

F is the number of faces of a solid

V is the number of vertices of a solid

E is the number of edges of a solid

For example, we know that a cube has 6 faces, 12 edges and 8 vertices. So,

Here F = 6, E = 12, and V = 8

Hence L. H. S., F + V – E = 6 + 8 - 12

= 14 – 12

= 2 (R. H. S.)

Hence, proved.

Let’s take one more example of a tetrahedron. It has 4 faces, 4 vertices and 6 edges. So,

Here F = 4, E = 6, and V = 4

Hence L. H. S., F + V – E = 4 + 4 – 6

= 8 – 6

= 2 (R. H. S.)

Hence, proved.

### Conclusion

In this article, we discussed basic properties like vertices, faces, and edges of 3-Dimensional shapes. We also discussed the same with different examples and learnt about Euler’s formula. We hope this article gave you required knowledge and information you need.