MATHEMATICMATHEMATICSS

Welcome To

R.R.Rozindar (Maths teacher)

Govt High School Hanjagi, Tq: Indi Dist: Bijapur.

PolyhedraPolyhedraClass X

OBJECTIVESOBJECTIVESAfter learning this chapter you will be able to• Define polygon, regular polygon and

polyhedra.• Identify different types of polyhedron.• Name different types of platonic solids.• Find the number of faces, vertices and

edges of given polyhedron.• Verify EULER’s formula.

IntroductionIntroductionYou may come across many solids of different

shapes such as

Sweet BoxCubical Eraser Gift Box Solar Cooker

Instrument BoxBoxTriangular prism

Prisms

IntroductionIntroduction

• What is the shape of each face of the objects you have seen ?

Ans : Ans : Triangle , Rectangle.Triangle , Rectangle.

• What these shapes usually called as ?

Ans : Ans : Polygons.Polygons.

PolyhedraPolyhedra

Definition of Polygon Regular polygon Polyhedron polyhedral solid Regular polyhedra

PolygonPolygon

• A closed figure bounded by straight line segments.

• The region bounded by polygon is polygonal region.

This is a This is a Polygonal Polygonal

RegionRegion

Regular PolygonRegular Polygon

• It is a polygon having equal sides and equal angles.

Equilateral Triangle

Square

PolyhedronPolyhedron

• A closed figure in the space bounded by polygonal faces is a polyhedron.

• A polyhedron divides the space in two regions, within the polyhedron and outside it.

Inner Region Of

Polyhedron

Polyhedral solidsPolyhedral solids

• A polyhedron has Edges, Vertices and Faces

Edges = 12Edges = 12

All sides are equalVertices = Vertices = 88

Faces =Faces =

11

443322

55

6666

Regular PolyhedraRegular Polyhedra• A polyhedron is called a regular polyhedron, if its

faces are congruent regular polygons. There are only Five regular polyhedrons. Tetrahedron

Hexahedron

Octahedron

Dodecahedron

Icosahedron

Regular PolyhedraRegular Polyhedra

• These polyhedra are known as PLATONIC SOLIDS

• It is believed that he associated these with

Fire Earth Air Universe Water

TetrahedronTetrahedron

• This polyhedron is called Tetrahedron because it has Four faces.

4

132

Faces = F = 4

TetrahedronTetrahedron• This polyhedron is called Tetrahedron

because it has Four faces.

Edges =

E = 6

Faces = F = 4

All sides are equal

TetrahedronTetrahedron• This polyhedron is called Tetrahedron

because it has Four faces.

• Faces = 4

• Edges = 6

Vertices = V

V = 4

HexahedronHexahedron• This polyhedron is called hexahedron

because it has six faces

1

654

3

2

Faces = F

F = 6

HexahedronHexahedron• This polyhedron is called hexahedron

because it has six faces

Faces =6

Edges = E

E = 12

+ +

HexahedronHexahedron• This polyhedron is called hexahedron

because it has six faces

Faces = 6

Edges = 12

Vertices = V

+V =

V = 8

OctahedronOctahedron• This polyhedron is called Octahedron

because it has Eight faces

Faces = 8

2233

44

88

55

11

6677

OctahedronOctahedron• This polyhedron is called Octahedron

because it has Eight faces

Faces = 8

Edges = E

E = 12

++ ++

OctahedronOctahedron• This polyhedron is called Octahedron

because it has Eight faces

Faces = 8

Edges = 12

+

V = 6

Vertices = V

DodecahedronDodecahedron• This polyhedron is called Dodecahedron because it has twelve

faces. • Number of Faces = F

2

3 5

1

6

4

11

712

10

9

8

Faces = F = 12

DodecahedronDodecahedron• This polyhedron is called Dodecahedron because it

has twelve faces. • Number of Faces=F=12

+ ++++

Edges = E

E = 30

DodecahedronDodecahedron• This polyhedron is called Dodecahedron

because it has twelve faces. • Number of Faces=F=12 Edges=E=30

++ +

Vertices = V

V = 20

IcosahedronIcosahedron• This polyhedron is called Icosahedron

because it has Twenty faces

1 2 3 4 5

897

106 1112

1314

15

16 17 18 19 20

Faces = F

F = 20

IcosahedronIcosahedron• This polyhedron is called Icosahedron

because it has Twenty faces

Faces = 20Edges = E

E = 30

= + +

+ +

+

IcosahedronIcosahedron• This polyhedron is called Icosahedron

because it has Twenty faces

Faces = 20Edges = E

Vertices = V

++ ++==

V= 12V= 12

Euler’s FormulaEuler’s FormulaSl. No

Name of the polyhedron

Faces Vertices Edges

11 Tetrahedron 44 44 66

22 Hexahedron 66 88 1212

33 Octahedron 88 66 1212

44 Dodecahedron 1212 2020 3030

55 Icosahedron 2020 1212 3030

F+VF+V E+2E+2

88 88

1414 1414

1414 1414

3232 3232

3232 3232

F + V = E + 2F + V = E + 2

Euler’s FormulaEuler’s Formula

• Euler’s formula for Euler’s formula for polyhedra is,polyhedra is,

• F + V = E + 2F + V = E + 2

Leonard Euler (1707-1783)

Test Your selfTest Your self

• Regular polyhedra are also called as ………………

• Shape of each face of tetrahedron is……………….

• Euler’s formula for polyhedron is ……………..

Platonic solidsPlatonic solids..

Equilateral TriangleEquilateral Triangle

F + V = E + 2.F + V = E + 2.

Home AssignmentHome Assignment

• Verify Euler’s formula for square based pyramid.

RecapRecap• Polygon:Polygon: -A closed figure bounded by straight line segments.• Regular polygon:Regular polygon: - Polygon having equal sides and equal angles.• Polyhedron:Polyhedron: - A closed figure in the space bounded by polygonal

faces.• Regular polyhedron:Regular polyhedron: - A polyhedron having congruent regular polygonal

faces. Platonic Solids:Platonic Solids: Tetrahedron. Hexahedron. Octahedron.

Dodecahedron. Icosahedron.

Know About.Know About.

• PLATOPLATO a Greek mathematician

• He founded the academy and over the entrance he inscribed, “ Let no one enter “ Let no one enter who is ignorant of who is ignorant of geometry”.geometry”.

Know About.Know About.University Entrance.University Entrance.

Know About.Know About.Ruins Of UniversityRuins Of University