{"id":143936,"date":"2022-04-17T10:30:34","date_gmt":"2022-04-17T05:00:34","guid":{"rendered":"https:\/\/www.aakash.ac.in\/blog\/?p=143936"},"modified":"2023-04-02T17:57:39","modified_gmt":"2023-04-02T12:27:39","slug":"vertices-faces-and-edges-cbse-12th-maths-concept","status":"publish","type":"post","link":"https:\/\/www.aakash.ac.in\/blog\/vertices-faces-and-edges-cbse-12th-maths-concept\/","title":{"rendered":"Vertices, Faces, and Edges: CBSE 12th Maths Concept"},"content":{"rendered":"

Objects have diverse physical characteristics that distinguish them: length, width, and diameter being a few of them. However, regardless of how different their dimensions are, they all take up space and have three dimensions. As a result, they are referred to as solids or three-dimensional shapes.<\/span><\/p>\n

Figures with two dimensions – length and breadth – can be depicted on a plane (for instance, a piece of paper). They are often referred to as two – dimensional figures or planar shapes.<\/span><\/p>\n

The three qualities that define any three-dimensional solid are vertices, faces, and edges.\u00a0<\/span><\/p>\n

Knowledge of these entities helps us define a solid accurately.\u00a0<\/span><\/p>\n

This article will explore the definitions of faces, edges, and vertices.<\/a> Along with that, we would also cover Euler’s formula and its characteristics. In the end, there are some solved examples and questions to polish and embed your mathematical concepts.<\/span><\/p>\n

What are Vertices?<\/h3>\n

Vertex ( plural vertices ) refers to the topmost point of a geometric figure. Since it is the highest point of a shape or a solid, it is also the farthest away from the figure’s base.\u00a0<\/span><\/p>\n

Another definition of a vertex is the point of meeting two sides of a plane figure. It can also be defined as a point of a\u00a0 geometrical figure that is the same for three or more sides. Every figure or shape made up of line segments has a set of vertices within it. They are the starting and ending points of edges too.<\/span><\/p>\n

Here is a list of various polyhedrons with their corresponding number of vertices:<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Polyhedron\u2019s name<\/b><\/td>\nNumber of Vertices<\/b><\/td>\n<\/tr>\n
Cube<\/span><\/td>\n8 Vertices<\/span><\/td>\n<\/tr>\n
Cone<\/span><\/td>\n1 vertex<\/span><\/td>\n<\/tr>\n
Sphere<\/span><\/td>\n0 vertices<\/span><\/td>\n<\/tr>\n
Cylinder<\/span><\/td>\n0 vertices<\/span><\/td>\n<\/tr>\n
Rectangular Prism<\/span><\/td>\n 8 vertices<\/span><\/td>\n<\/tr>\n
Triangular Prism<\/span><\/td>\n6 vertices<\/span><\/td>\n<\/tr>\n
Hexagonal Prism<\/span><\/td>\n12 vertices<\/span><\/td>\n<\/tr>\n
Pentagonal Prism<\/span><\/td>\n10 vertices<\/span><\/td>\n<\/tr>\n
Square pyramid<\/span><\/td>\n5 vertices<\/span><\/td>\n<\/tr>\n
Octagonal prism<\/span><\/td>\n16 vertices<\/span><\/td>\n<\/tr>\n
Triangular pyramid<\/span><\/td>\n4 vertices<\/span><\/td>\n<\/tr>\n
Rectangular pyramid<\/span><\/td>\n5 vertices<\/span><\/td>\n<\/tr>\n
Hexagonal pyramid\u00a0<\/span><\/td>\n6 vertices<\/span><\/td>\n<\/tr>\n
Octagonal Pyramid<\/span><\/td>\n7 vertices<\/span><\/td>\n<\/tr>\n
Tetrahedron<\/span><\/td>\n9 vertices<\/span><\/td>\n<\/tr>\n
octahedron<\/span><\/td>\n6 vertices<\/span><\/td>\n<\/tr>\n
Icosahedron<\/span><\/td>\n12 vertices<\/span><\/td>\n<\/tr>\n
Dodecahedron<\/span><\/td>\n20 vertices<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

What are Edges?<\/h3>\n

Edges of a figure are referred to as the outline of a shape or a solid. They can also be defined as the set of lines that join together to form vertices of various figures.<\/span><\/p>\n

When we talk about the relation between edges and vertices of a figure, we must acknowledge that there exist considerable contrasts between two-dimensional and three-dimensional figures.\u00a0<\/span><\/p>\n

For instance, in 2-D figures like squares and triangles – there are four edges and four vertices; and three edges and three vertices, respectively. However, in the case of a 3-D solid like a cube – there are 12 edges and 8 vertices. Hence, even though the edges and vertices can be identical in number in the case of 2-D shapes, it is not the case when we talk about 3-D figures.<\/span><\/p>\n

Here is a list of various polyhedrons with their corresponding number of edges:<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Polyhedron\u2019s name<\/span><\/td>\nNumber of edges<\/span><\/td>\n<\/tr>\n
Cube<\/span><\/td>\n12 edges<\/span><\/td>\n<\/tr>\n
Cone<\/span><\/td>\n1 edges<\/span><\/td>\n<\/tr>\n
Sphere<\/span><\/td>\n0 edges<\/span><\/td>\n<\/tr>\n
Cylinder<\/span><\/td>\n2 edges<\/span><\/td>\n<\/tr>\n
Rectangular Prism<\/span><\/td>\n12 edges<\/span><\/td>\n<\/tr>\n
Triangular Prism<\/span><\/td>\n9 edges<\/span><\/td>\n<\/tr>\n
Hexagonal Prism<\/span><\/td>\n18 edges<\/span><\/td>\n<\/tr>\n
Pentagonal Prism<\/span><\/td>\n15 edges<\/span><\/td>\n<\/tr>\n
Square pyramid<\/span><\/td>\n8 edges<\/span><\/td>\n<\/tr>\n
Octagonal prism<\/span><\/td>\n24 edges<\/span><\/td>\n<\/tr>\n
Triangular pyramid<\/span><\/td>\n6 edges<\/span><\/td>\n<\/tr>\n
Rectangular pyramid<\/span><\/td>\n8 edges<\/span><\/td>\n<\/tr>\n
Hexagonal pyramid\u00a0<\/span><\/td>\n12 edges<\/span><\/td>\n<\/tr>\n
Octagonal Pyramid<\/span><\/td>\n16 edges<\/span><\/td>\n<\/tr>\n
Tetrahedron<\/span><\/td>\n6 edges<\/span><\/td>\n<\/tr>\n
octahedron<\/span><\/td>\n12 edges<\/span><\/td>\n<\/tr>\n
Icosahedron<\/span><\/td>\n30 edges<\/span><\/td>\n<\/tr>\n
Dodecahedron<\/span><\/td>\n30 edges<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

What are faces?<\/h3>\n

One of the essential components of a geometrical figure, the face, is defined as any specific flat surface of a solid object.\u00a0<\/span><\/p>\n

The face of a 3-D figure is usually a 2-D shape. For instance, the faces of a cube would be in the shape of squares. The exception to this rule is a sphere with only one face with no vertices or edges.<\/span><\/p>\n

The faces of 3-D figures can be of various kinds – ranging from curved faces to flat faces. We often observe different 3-D figures that comprise a combination of these kinds of surfaces. For example, a cone (3-D figure) has one flat face and one curved face.<\/span><\/p>\n

Here is a list of polyhedrons with their corresponding number of faces:<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Polyhedron\u2019s name<\/span><\/td>\nNumber of Faces<\/span><\/td>\n<\/tr>\n
Cube<\/span><\/td>\n6 faces<\/span><\/td>\n<\/tr>\n
Cone<\/span><\/td>\n2 faces (1 flat face and 1 curved face)<\/span><\/td>\n<\/tr>\n
Sphere<\/span><\/td>\n1 continuous face<\/span><\/td>\n<\/tr>\n
Cylinder<\/span><\/td>\n3 faces (2 flat faces and 1 curved face)<\/span><\/td>\n<\/tr>\n
Rectangular prism<\/span><\/td>\n6 faces<\/span><\/td>\n<\/tr>\n
Triangular prism<\/span><\/td>\n5 faces<\/span><\/td>\n<\/tr>\n
Hexagonal prism<\/span><\/td>\n8 faces<\/span><\/td>\n<\/tr>\n
Pentagonal prism<\/span><\/td>\n7 faces<\/span><\/td>\n<\/tr>\n
Square pyramid<\/span><\/td>\n5 faces<\/span><\/td>\n<\/tr>\n
Octagonal prism<\/span><\/td>\n10 faces<\/span><\/td>\n<\/tr>\n
Triangular pyramid<\/span><\/td>\n4 faces<\/span><\/td>\n<\/tr>\n
Rectangular pyramid<\/span><\/td>\n5 faces<\/span><\/td>\n<\/tr>\n
Hexagonal pyramid\u00a0<\/span><\/td>\n7 faces<\/span><\/td>\n<\/tr>\n
Octagonal pyramid<\/span><\/td>\n9 faces<\/span><\/td>\n<\/tr>\n
Tetrahedron<\/span><\/td>\n4 faces<\/span><\/td>\n<\/tr>\n
Octahedron<\/span><\/td>\n8 faces<\/span><\/td>\n<\/tr>\n
Icosahedron<\/span><\/td>\n20 faces<\/span><\/td>\n<\/tr>\n
Dodecahedron<\/span><\/td>\n10 faces<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Euler’s formula<\/h3>\n

Now that we are clear with what vertices, edges, and faces are – we can now move on to how they are related. Using Euler’s formula, we calculate the relationship between various geometrical figures’ vertices, faces, and edges.<\/span><\/p>\n

It’s important to remember that the formula only applies to closed solids with flat sides and straight edges and is irrelevant for figures with curved edges. For instance, we can undoubtedly use Euler’s formula for cubes and cuboids, but it is not used in the case of cylinders.<\/span><\/p>\n

The Euler theorem, named after Leonhard Euler, is one of the most important mathematical theorems. This theorem helps us calculate and analyze various components of a 3-D geometrical figure. Using this formula, we need not count the edges, vertices, and faces of a figure every time we need the numbers.\u00a0<\/span><\/p>\n

According to Euler’s theorem,\u00a0<\/span><\/p>\n

F+V-E = 2, where F represents the number of faces, V represents the number of vertices, and E represents the number of edges.\u00a0<\/span><\/p>\n

Let us now take a real-life example and apply Euler’s formula ourselves.<\/span><\/p>\n

We will apply Euler’s formula to a tetrahedron, a 3-D geometrical figure with no curved surface.<\/span><\/p>\n\n\n\n\n\n\n
Entities of tetrahedron<\/b><\/td>\nTetrahedron<\/b><\/td>\n<\/tr>\n
Number of Faces<\/span><\/td>\n4<\/span><\/td>\n<\/tr>\n
Number of Vertices<\/span><\/td>\n4<\/span><\/td>\n<\/tr>\n
Number of Edges<\/span><\/td>\n6<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Now according to the Euler’s formula, we have:<\/span><\/p>\n

F+V-E = 2 (where F = faces, V= vertices, and E= edges)<\/span><\/p>\n

Hence,<\/span><\/p>\n

4+4-6 =2<\/span><\/p>\n

8-6 = 2<\/span><\/p>\n

2 = 2<\/span><\/p>\n

Since LHS=RHS, we can say that a tetrahedron follows Euler’s formula. Hence, this theorem is used to find out any one of the three missing entities of the figure.<\/span><\/p>\n

Euler’s characteristics<\/b><\/p>\n

F+V-E = sigma, which is Euler’s characteristic of being any integer value<\/span><\/p>\n

But specifically for convex polygons, the value is considered as 2.<\/span><\/p>\n

Solved Examples<\/b><\/p>\n

    \n
  1. Find the number of faces in a polyhedron that has 8 vertices and 12 edges. What is the shape you obtain?<\/span><\/li>\n
  2. According to the Euler\u2019s formula, we get\u00a0<\/span><\/li>\n<\/ol>\n

    F+V-E = 2<\/span><\/p>\n

    Rewriting this, we get:<\/span><\/p>\n

    F= E-V+2\u00a0<\/span><\/p>\n

    F= 12-8+2<\/span><\/p>\n

    F= 4+2<\/span><\/p>\n

    F=6.<\/span><\/p>\n

    The polyhedron thus obtained with 6 faces is a cube.<\/span><\/p>\n

      \n
    1. Prove the Euler\u2019s formula for a prism?<\/span><\/li>\n
    2. According to the Formula, F+V-E = 2.<\/span><\/li>\n<\/ol>\n

      Faces in a prism = 5<\/span><\/p>\n

      Edges in a prism = 9<\/span><\/p>\n

      Vertices in a prism = 6<\/span><\/p>\n

      Substituting the value in the formula, we get:<\/span><\/p>\n

      5+6-9=2<\/span><\/p>\n

      11-9 = 2<\/span><\/p>\n

      2=2<\/span><\/p>\n

      Hence proved.<\/span><\/p>\n

        \n
      1. Can a polyhedron have 20 faces, 10 edges, and 15 vertices?<\/span><\/li>\n
      2. If a polyhedron has 20 faces, 10 edges, and 15 vertices – it should be able to satisfy Euler\u2019s formula.<\/span><\/li>\n<\/ol>\n

        F+V-E=2<\/span><\/p>\n

        Here, the number of faces F = 20<\/span><\/p>\n

        Number of edges E = 10<\/span><\/p>\n

        Number of vertices V = 15<\/span><\/p>\n

        Hence, by Euler’s formula:<\/span><\/p>\n

        LHS: 20+15 = 35<\/span><\/p>\n

        RHS:\u00a0 10+2 = 12,<\/span><\/p>\n

        which is not true because 35\u226012<\/span><\/p>\n

        Hence, Euler’s formula is not satisfied. This proves that such a polyhedron does not exist.<\/span><\/p>\n

        Questions to Polish your Concepts<\/b><\/p>\n

          \n
        1. Prove Euler\u2019s formula for octahedra.<\/span><\/li>\n
        2. Using Euler’s formula find the unknown:<\/span><\/li>\n<\/ol>\n\n\n\n\n\n
          Faces<\/span><\/td>\n?<\/span><\/td>\n16<\/span><\/td>\n20<\/span><\/td>\n<\/tr>\n
          Vertices<\/span><\/td>\n20<\/span><\/td>\n?<\/span><\/td>\n12<\/span><\/td>\n<\/tr>\n
          Edges<\/span><\/td>\n30<\/span><\/td>\n24<\/span><\/td>\n?<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          3. Can a polyhedron have for its faces:<\/span><\/p>\n

          (i) 2 triangles?<\/span><\/p>\n

          (ii) 5 triangles?<\/span><\/p>\n

          (iii) a square and three triangles?<\/span><\/p>\n

          4. Is a square prism and a cube the same thing?<\/p>\n

          Conclusion<\/h3>\n

          We come across objects of various sizes and shapes in our day-to-day lives, and these objects are nothing but polyhedrons with fixed dimensions and properties. Properties such as Faces, Edges, and vertices. A vertex is the intersection of two or more line segments.\u00a0<\/span><\/p>\n

          A face of a figure is usually defined as any individual flat surface of a solid object, and a point where two faces meet can be characterized as an edge. All three entities are linked to each other, and the relation between them is given by F+V-E=2, which is Euler’s formula.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

          Objects have diverse physical characteristics that distinguish them: length, width, and diameter being a few of them. However, regardless of how different their dimensions are, they all take up space and have three dimensions. As a result, they are referred to as solids or three-dimensional shapes. 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