Faces, Edges And Vertices of 3D Shapes (2024)

Faces, Edges, And Vertices of 3D Shapes: Faces, Edges, and Vertices are the three key components that are used to define various 3D objects. They have different dimensions like length, width, and height.

In geometry, Faces are the flat surfaces of a 3D shape. They are bounded by edges and are what give the shape its appearance. Edges are the straight lines where two faces of a 3D shape meet. They form the boundaries between faces and help define the shape’s overall structure. Vertices (singular: vertex) are the points where the edges of a 3D shape meet. They are essentially the corners of the shape.

In this article, we are going to learn about the faces, edges, and vertices of different 3D shapes in detail.

What are Faces?

A face of a shape is defined as the flat surface of a solid 3D Shape. It can be referred to as the outer surface of a solid object, whether it be a curved face or a straight face.

The figure given below is a cuboid that has six faces, i.e., ABCD, EFGH, ADHE, AEFB, BFGC, and DHGC.

What are Edges?

An edge is defined as the line segment where the faces of a solid meet. Even though many shapes have straight edges and lines, some shapes, like hemispheres, have curved edges.

What are Vertices?

In geometry, a vertices are point of intersection of two or more curves, lines, or edges. They are represented as points and denoted by letters such as A, B, C, D, etc.

The figure given below is a cuboid that has eight vertices, i.e., A, B, C, D, E, F, G, and H.

What are Polyhedrons?

Polyhedron is referred to as a three-dimensional solid that is made up of polygons. It consists of flat polygonal faces, straight edges, and sharp corners called vertices.

Some examples of polyhedrons are cubes, pyramids, prisms, etc.

Types of Polyhedrons

Different types of polyhedrons have been discussed in the table below:

Note : As cones, cylinders, and spheres do not have polygonal faces and curved surfaces, they are non-polyhedrons.

Faces, Edges And Vertices of 3D Shapes

Now we will find the vertices, faces and edges of various solid shapes with the help of a table given below.

Shape	Figure	Vertices	Edges	Faces
Cube		8	12	6
Cuboid		8	12	6
Cone		1	1	2
Cylinder			2	3
Sphere				1
Triangular Prism		6	9	5
Rectangular Prism		8	12	6
Square Prism		8	12	6
Pentagonal Prism		10	15	7
Hexagonal Prism		12	18	8
Triangular Pyramid		4	6	4
Rectangular Pyramid		5	8	5
Square Pyramid		5	8	5
Pentagonal Pyramid		6	10	6
Hexagonal Pyramid		7	12	7

Relation Between Faces, Edges And Vertices of 3D Shapes

The relationship between vertices, faces, and edges can be determined using Euler’s formula. Euler’s formula states that for any convex polyhedron, the sum of the number of faces (F) and vertices (V) is exactly two greater than the number of edges (E).

F + V = 2 + E

where,

F is the number of faces,
V is the number of vertices,
E is the number of edges

Euler’s formula is applicable for closed solids that have flat sides and straight edges, such as cuboids. It is not applied to solids that have curved edges, such as cylinders, spheres, etc.

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Faces, Edges And Vertices of 3D Shapes Examples

Example 1: Verify Euler’s formula for the heptahedron.

Solution:

A heptahedron has 7 faces, 10 vertices, and 15 edges.

From Euler’s formula, we have,

F + V = 2 + E

⇒ 7 + 10 = 2 + 15

⇒ 17 = 17

Hence, Euler’s formula is verified for the heptahedron.

Example 2: Calculate the number of edges of a polyhedron if it has 6 vertices and 9 faces.

Solution:

Given data,

V = 6 and F = 5

From Euler’s formula, we have,

F + V = 2 + E

⇒ 5 + 6 = 2 + E

⇒ 11 = 2 + E

⇒ E = 11 – 2 = 9

Hence, the given polyhedron has 9 edges.

Example 3: Calculate the number of vertices of a polyhedron if it has 5 faces and 8 edges.

Solution:

Given data,

F = 5 and E = 8

From Euler’s formula, we have,

F + V = 2 + E

⇒ 5 + V = 2 + 8

⇒ 5 + V = 10

⇒ V = 10 – 5 = 5

Hence, the given polyhedron has 5 vertices.

Conclusion

Understanding the faces, edges, and vertices of 3D shapes is key to understand the basics of geometrical figures. These elements help define the structure and properties of three-dimensional objects, from cubes and pyramids to spheres and cylinders. Knowing how to count faces, edges, and vertices allows us to better visualize and work with shapes in both math and real-life situations.

Faces, Edges And Vertices of 3D Shapes -FAQs

What is polyhedron?

In geometry, a polyhedron is referred to as a three-dimensional solid that is made up of polygons. A polyhedron consists of flat polygonal faces, straight edges, and sharp corners called vertices. Some examples of polyhedrons are cubes, pyramids, prisms, etc.

What are Vertices, Faces and Edges?

A vertex (plural: vertices) is a point where two or more lines meet. An edge is the line segment that connects two vertices, forming the skeleton or outline of a shape. A face is a flat surface enclosed by edges. In a three-dimensional object, such as a cube, the vertices are the corners, the edges are the lines connecting these corners, and the faces are the flat surfaces bounded by these edges.

How Many Vertices, Faces and Edges does a Cone have?

A cone has 1 vertex, 2 faces, and 1 edge.

How Many Vertices, Faces and Edges does a Cylinder have?

A cylinder has 0 vertices, 3 faces, and 2 edges.

How Many Vertices, Faces and Edges does a Cube have?

A cube has 8 vertices, 6 faces, and 12 edges.

How Many Vertices, Faces and Edges does a Cuboid have?

A cuboid has 8 vertices, 6 faces, and 12 edges.

What is a face of a solid?

A face is defined as the flat surface of a solid. It can be referred to as the outer surface of a solid object, whether it be a curved face or a straight face. For example, a cube has six faces, a tetrahedron has four faces, a cylinder has three faces, i.e., a curved face and two circular faces, a cone has two faces, i.e., a curved face and a flat face, whereas a sphere has a curved surface.

State Euler’s Formula.

The relationship between vertices, faces, and edges can be determined using Euler’s formula. Euler’s formula states that for any convex polyhedron, the sum of the number of faces (F) and vertices (V) is exactly two greater than the number of edges (E).

Euler’s Formula:

F + V = 2 + E

where,
F is the number of faces,
V is the number of vertices,
E is the number of edges