POLYHEDRONS AND NON-POLYHEDRONS SHAPES

Visualising solid shapes of Class 8

A polyhedron is a geometric solid in three dimensions with flat faces and straight edges.

POLYHEDRONS AND NON-POLYHEDRONSCuboid.gifrectangle

POLYHEDRONS AND NON-POLYHEDRONS Hexagon

Traingular Prism A Regular Polyhedron

Polyhedral Surface

A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.

Edges

Edges have two important characteristics (unless the polyhedron is complex):

  • An edge joins just two vertices.
  • An edge joins just two faces.

These two characteristics are dual to each other.

Common Polyhedra

POLYHEDRONS AND NON-POLYHEDRONSplatonic platonicPlatonicPlatonic

Platonic Solids

POLYHEDRONS AND NON-POLYHEDRONSPrismsPrisms

Prisms

POLYHEDRONS AND NON-POLYHEDRONSPyramidsPyramids

Pyramids

Convex Polyhedrons

The idea of convex polyhedron is similar to that of convex polygons. 

These are convex

POLYHEDRONS AND NON-POLYHEDRONS

polyhedrons

POLYHEDRONS AND NON-POLYHEDRONS

These are not convex polyhedrons

Regular Polyhedrons

A polyhedron is said to be regular if its faces are made up of regular polygons and the same number of faces meet at each vertex. 

POLYHEDRONS AND NON-POLYHEDRONS

1. This polyhedron is regular.

2. Its faces are congruent, regular polygons. Vertices are formed by the same number of faces

POLYHEDRONS AND NON-POLYHEDRONS

1. This polyhedron is not regular.

2. All the sides are congruent; but the vertices are not formed by the same number of faces. 3 faces meet at A but 4 faces meet at B.

Two important members of polyhedron family around are prisms and pyramids.

POLYHEDRONS AND NON-POLYHEDRONS

These are prisms

 

POLYHEDRONS AND NON-POLYHEDRONS

These are pyramids

Prism

A solid whose two faces are parallel plane polygons and the side faces are rectangles is called a prism. A solid whose base and top are identical polygons and the sides are rectangles, is known as a prism. It is a polyhedron, two of whose faces are congruent polygons in parallel planes and whose other faces are parallelograms.

TYPES

FIGURE

FACES

EDGES

VERTICES

(i)

Triangular Prism

POLYHEDRONS AND NON-POLYHEDRONS

5

9

6

(ii)

Cuboid Rectangular Prism

POLYHEDRONS AND NON-POLYHEDRONS

6

12

8

(iii)

Square Prism

POLYHEDRONS AND NON-POLYHEDRONS

6

12

8

(iv)

Cube

POLYHEDRONS AND NON-POLYHEDRONS

6

12

8

 

(v)

 

Pentagonal Prism

POLYHEDRONS AND NON-POLYHEDRONS

 

7

 

15

 

10

 

(vi)

 

Cylinder

POLYHEDRONS AND NON-POLYHEDRONS

 

3

 

2

 

Pyramids

A pyramid is a polyhedron whose base is a polygon (of any number of sides) and whose other faces are triangles with a common vertex.


TYPES

FIGURE

FACES

EDGES

VERTICES

(i)

Triangular Pyramid

POLYHEDRONS AND NON-POLYHEDRONS

4

6

4

(ii)

Rectangular Pyramid

POLYHEDRONS AND NON-POLYHEDRONS

5

8

5

(iii)

Square Pyramid

POLYHEDRONS AND NON-POLYHEDRONS

5

8

5

(iv)

Pentagonal Pyramid

POLYHEDRONS AND NON-POLYHEDRONS

6

10

6

  • A prism is a polyhedron whose base and top are congruent polygons and whose other faces, i.e., lateral faces are parallelograms in shape. 
  • A pyramid is a polyhedron whose base is a polygon (of any number of sides) and whose lateral faces are triangles with a common vertex.
  • A prism or a pyramid is named after its base. Thus a hexagonal prism has a hexagon as its base; and a triangular pyramid has a triangle as its base.

Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:

3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them.

2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface.

1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton.

0 dimensions: A vertex (plural vertices) is a corner point.

-1 dimension: The null polytope is a kind of non-entity required by abstract theories.

EULER’S Formula

The table below shows the number of faces, edges and vertices of each of the platonic solids. Here, v stands for vertices, f for faces and e for edges.

Solid

F

V

E

F + V

E + 2

Hexahedron (Cube)

Octahedron

Dodecahedron

Icosohedron

6

8

12

20

8

6

20

12

12

12

30

30

6 + 8 = 14

8 + 6 = 14

12 + 20 + 32

20 + 12 = 32

12 + 2 = 14

12 + 2 = 14

30 + 2 = 32

30 + 2 = 32

The above table clearly shows that

F + V = E + 2

Leonard Euler (1707-1783) discovered this formula which established the relationship among the number of faces, edges and vertices of a polyhedron.

Euler’s Formula

F + V = E + 2

Where F = number of faces

V = number of vertices

E = number of edges.

e.g. Try it on the cube:

POLYHEDRONS AND NON-POLYHEDRONS

A cube has 6 Faces, 8 Vertices, and 12 Edges,

so: 6 + 8 - 12 = 2

EULER CHARACTERISTIC:

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

x = V – E + F

For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected and whose boundary is a manifold, χ = 2.

So, F+V-E can equal 2, or 1, and maybe other values, so the more general formula is

F + V - E = χ

Where χ is called the "Euler Characteristic".

Here are a few examples:

Shape

 

χ

Sphere

POLYHEDRONS AND NON-POLYHEDRONS

2

Torus

POLYHEDRONS AND NON-POLYHEDRONS

0

Mobius Strip

POLYHEDRONS AND NON-POLYHEDRONS

0

POLYHEDRONS AND NON-POLYHEDRONS

And the Euler Characteristic can also be less than zero.

This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so:

F + V - E = -2

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