Faces, Edges and Vertices: Relationship and Examples

In our daily lives, we come across a variety of objects of various forms and sizes. The majority of these objects have some length, breadth, and height in common. One-dimensional things are those that only have length. A one-dimensional figure is a line. Faces, Edges, and Vertices will be discussed in-depth in this article.

Two-dimensional shapes such as rectangles, triangles, and circles have length and breadth. Three-dimensional things are those that have length, width, and height. Three-dimensional objects include cubes, cuboids, cones, and so on. Continue reading to learn about the definitions of faces, edges, and vertices, as well as the number of faces, edges, and vertices found in various three-dimensional objects.

Three-Dimensional Objects

\(3\,D\) shapes are solid shapes or objects with three dimensions, i.e. length, width, and height. Faces, edges, and vertices are other significant concepts connected with \(3\,D\) shapes. They have depth; therefore, they take up some space. The base and top portions of cross-sections of several \(3\,D\) objects are \(2\,D\) shapes.

A soccer ball, a cube, a bucket, and a book are some real-life examples of \(3\,D\) shapes displayed below.

The most significant concepts connected with three-dimensional objects are faces, edges and vertices. This article will study the definition of faces, edges and vertices and the number of faces, edges, and vertices of some three-dimensional objects.

Faces

Every flat surface of a solid is known as the face of that solid. In other words, the surfaces on the outside of a shape are known as the face of that shape.

Edges

Any two adjacent faces of a solid meet in a line segment called an edge of that solid. In other words, an edge is a line segment that serves as an interface between two faces.

Vertices

For any two edges that meet at an end-point, a third edge also meets them. This point of intersection of three edges of a solid is called a vertex of the solid. In other words, a point where two or more edges meet is called a vertex.

Polyhedron

A \(3\,D\) shape or object is made up of a variety of components. Polygonal regions make up the majority of the solid figures. Faces, edges, and vertices are the three types of regions. Polyhedrons are solid geometric shapes that have faces, edges, and vertices.

There are two types of the polyhedron: concave polyhedron and convex polyhedron

Convex Polyhedron: A convex polyhedron is one in which the polyhedron’s surface (which comprises its faces, edges, and vertices) does not intersect, and the line segment joins any two points of the polyhedron lies within its interior part or surface.

Concave Polyhedron: A polyhedron that is not a convex polyhedron is concave.

Relationship Between Faces, Edges and Vertices – Euler’s Formula

Euler’s formula can be used to calculate the relationship between vertices, faces, and edges. Let us take notice of an exciting link between solids, faces, edges, and vertices now that we’ve learnt about them.

It’s important to remember that the formula only applies to closed solids with flat sides and straight edges, such as cuboids. Because cylinders have curved edges, they cannot be used.
\(F + V – E = 2\) gives Euler’s formula.
Where \(F\) is the number of faces, \(V\) is the number of vertices, and \(E\) is the number of edges of the given solid.
So, from Euler’s formula, we can say that the number of faces \(\left( F \right)\) and vertices \(\left( V \right)\) added together for each convex polyhedron is exactly two more than the number of edges \(\left( E \right).\)

Faces, Edges and Vertices of a Cube

A cube is a symmetrical three-dimensional solid, having six square faces.

A cube has \(6\) square faces, \(12\) edges and \(8\) vertices.
Therefore, \(F + V – E = 2 \Rightarrow 6 + 8 – 12 = 2.\)

Faces, Edges and Vertices of a Cuboid

A cuboid is a three-dimensional solid bounded by six rectangular plane regions.

A cuboid has \(6\) rectangular faces, \(12\) edges and \(8\) vertices.
Therefore, \(F + V – E = 2 \Rightarrow 6 + 8 – 12 = 2.\)

Learn How to Visualise Solid Shapes

Faces, Edges and Vertices of a Triangular Pyramid

A pyramid is a polyhedron with a base and three or more triangle faces that meet above the base at a point called the apex. A pyramid with a triangle base is known as a triangular pyramid.

A triangular pyramid has \(4\) faces, \(6\) edges and \(4\) vertices. Therefore, \(F + V – E = 2 \Rightarrow 4 + 4 – 6 = 2.\)

Faces, Edges and Vertices of a Cylinder

A cylinder is a three-dimensional solid that holds two parallel bases at a fixed distance linked by a curving surface. These bases are usually circular (like a circle), and a line segment called the axis connects the two bases’ centres. The height is the perpendicular distance between the bases.

A cylinder has two circular faces, one curved face, two curved edges and no vertex.

Faces, Edges and Vertices of a Cone

A cone is a shape created by connecting a common point, called the apex or vertex, to all the points of a circular base using a collection of line segments. The height of the cone is the distance between the vertex and the base. The slant height is the length of the cone from the apex to any point on the base’s perimeter.

A cone has one vertex, one curved edge, one curved face and one flat face. Therefore, \(F + V – E = 2 \Rightarrow 2 + 1 – 1 = 2.\)

Faces, Edges and Vertices of a Sphere

A sphere is a collection of points in three-dimensional space that are all at equal distances from a common point. The radius of a sphere is a constant distance, and the common point is the sphere’s centre.

A sphere has a curved surface, no edges, and no vertices.

Faces, Edges and Vertices of a Pentagonal Prism

A pentagonal prism is a three-dimensional solid with bottom and top pentagonal bases. A rectangle is the shape of all the other sides of a pentagonal prism.

A pentagonal prism has \(7\) faces, \(10\) vertices and \(15\) edges. Therefore, \(F + V – E = 2 \Rightarrow 7 + 10 – 15 = 2.\)

Faces, Edges and Vertices of a Square Pyramid

A pyramid is a polyhedron with a base and three or more triangle faces that meet above the base at a point (the apex). In the case of a square pyramid, the base has four sides and is a square.

A square pyramid has \(5\) faces, \(8\) edges and \(5\) vertices. Therefore, \(F + V – E = 2 \Rightarrow 5 + 5 – 8 = 2.\)

Faces, Edges and Vertices of a Tetrahedron

Tetrahedrons are three-dimensional polygons with four triangular faces. The base of a tetrahedron is one of the triangles, and the other three triangles combine to make the pyramid. The tetrahedron is a polyhedron with triangle faces linking the base to a common point and a flat polygon base, and it is one of the pyramid varieties.

A tetrahedron has \(4\) faces, \(4\) vertices and \(6\) edges. Therefore, \(F + V – E = 2 \Rightarrow 4 + 4 – 6 = 2.\)

Solved Examples – Faces, Edges and Vertices

Q.1. Verify Euler’s formula for the tetrahedron.
Ans: We have \(4\) faces, \(4\) vertices, and \(6\) edges in a tetrahedron. So, according to Euler’s formula, we have \(F + V – E = 2.\)
\( \Rightarrow 4 + 4 – 6 = 2\)
\( \Rightarrow 2 = 2\)
Therefore, Euler’s formula is verified for the tetrahedron.

Q.2. Verify Euler’s formula for the octahedron.
Ans: We know that we have \(8\) faces, \(6\) vertices, and \(12\) edges in an octahedron. So, according to Euler’s formula, we have \(F + V – E = 2.\)
\( \Rightarrow 8 + 6 = 2\)
\( \Rightarrow 2 = 2\)
Therefore, Euler’s formula is verified for the octahedron.

Q.3. Verify Euler’s formula for the dodecahedron.
Ans: We know that we have \(12\) faces, \(20\) vertices, and \(30\) edges in a dodecahedron. So, according to Euler’s formula, we have \(F + V – E = 2.\)
\( \Rightarrow 12 + 20 – 30 = 2\)
\( \Rightarrow 2 = 2\)
Therefore, Euler’s formula is verified for the dodecahedron.

Q.4. Ram knows that a polyhedron has 12 vertices and 30 edges. How can he find the number of faces?
Ans: We know that \(F + V – E = 2\) gives Euler’s formula for a polyhedron.
Given: \(V = 12\) and \(E = 30\)
Substituting the given values in Euler’s formula, we get
\(F + 12 – 30 = 2\)
\( \Rightarrow F = 32 – 12\)
\( \Rightarrow F = 20\)
Therefore, the number of faces of the given polyhedron is \(20.\)

Q.5. Can we have a polyhedron with 9 faces, 12 vertices and 18 edges?
Ans: We know that \(F + V – E = 2\) gives Euler’s formula for a polyhedron.
Substituting the given values in Euler’s formula, we get
\(9 + 12 – 18 = 3\)
\( \Rightarrow 3 \ne 2\)
Therefore, we cannot have a polyhedron with \(9\) faces, \(12\) vertices and \(18\) edges.

Summary

In this article, we learnt the meaning of three-dimensional objects, the definition of faces, edges and vertices and the number of faces, edges and vertices in a cube, cuboid, triangular pyramid, cylinder, cone, square pyramid, pentagonal prism, sphere. Also, we have learnt the relationship between faces, edges, and vertices called Euler’s formula and solved some example problems.

Learn All the Concepts on Euler’s Formula

Frequently Asked Questions

We have provided some frequently asked questions on Faces, Edges and Vertices here:

Q.1. How many vertices a cube has?
Ans: A cube has \(8\) vertices.

Q.2. How many edges do a cone has?
Ans: A cone has one curved edge.

Q.3. State Euler’s formula for a polyhedron.
Ans: The number of faces \(\left( F \right)\) and vertices \(\left( V \right)\) added together for each convex polyhedron is exactly two more than the number of edges \(\left( E \right).\)
\(F + V – E = 2\) gives Euler’s formula.

Q.4. What are faces, edges and vertices in solid figures?
Ans: Every flat surface of a solid is known as the face of that solid.
Any two adjacent faces of a solid meet in a line segment called an edge of that solid.
A point where two or more edges meet is called a vertex.

Q.5. Which shape has 5 faces, 6 vertices and 9 edges?
Ans: The solid shape with \(5\) faces, \(6\) vertices and \(9\) edges is a triangular prism.

Now you are provided with all the necessary information on the concept of faces, edges and vertices and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.