1. Liceo Scientifico Isaac Newton Maths course Polyhedra
Professor
Tiziana De Santis

2. A convex polyhedron is the part of space bounded by n polygons (with n ≥ 4) belonging to different planes, so that each edge of the polyhedron is the intersection of two of them
face
vertex
edge
diagonal

3. Euler’s relation F + V – E = 2
All convex polyhedra satisfy this important relation between the numbers of faces (F), of vertices (V) and of edges (E)
32 faces 20 12
90 edges
60 vertices
– 90 = 2

4. Regular polyhedra
A polyhedron is said to be regular if its faces are regular and congruent polygons, and its dihedral angles and solid angles are also congruent
These solids are also called platonic

5. Tetrahedron It has four faces, four vertices and six edges
Three equilateral triangles converge in each vertex
Symmetries:
- 6 planes passing through the barycentre containing one edge
- 3 lines passing through middle points of opposite edges
( Euler: – 6 = 2 )

6. Octahedron It has eight faces, six vertices and twelve edges
Four equilateral triangles converge in each vertex
Symmetries:
intersection of diagonals identify the symmetry centre
3 symmetry axes link opposite vertices
6 symmetry axes pass through middle points of parallel edges
9 symmetry planes, 3 of which pass through 4 parallel edges two by two and 6 passing through 2 opposite vertices and middle points of opposite edges
( Euler: – 12 = 2 )

7. Hexahedron It has six faces, eight vertices and twelve edges
Three squares converge in each vertex
Symmetries:
intersection of diagonals identifies the symmetry centre
9 symmetry axes: 3 passing through centres of opposite faces, 6 passing through middle points of opposite edges
9 symmetry planes (3 median planes and 6 diagonal planes)
( Euler: – 12 = 2)

8. Icosahedron Dodecahedron Some symmetries: it has a symmetry center,
axes passing through opposite vertices of opposite faces,
planes containing edges of opposite faces
It has twenty faces, twelve vertices and thirty edges
Five equilateral triangles converge in each vertex
(Euler: – 30 = 2)
Dodecahedron
Some symmetries: it has
a symmetry center,
planes passing through barycenter containing one edge
lines passing through opposite vertices of opposite faces
It has twelve faces, twenty vertices and thirty edges
Three pentagons converge in each vertex
(Euler: – 30 = 2)

9. A solid angle must have at least three faces
The sum of the angles of the faces must be less than 360°
360°
It is possible to demonstrate that
there are only five regular polyhedra

10. To construct a polyhedron with equilateral triangles:
3 faces converge at each vertex
3 x 60°=180°<360° (tetrahedron)
4 faces converge at each vertex
4 x 60°=240°<360° (octahedron)
5 faces converge at each vertex
5 x 60°=300°<360° (icosahedron)
It is impossible for 6 or more faces to converge in one vertex because:
6 x 60° = 360°

11. To construct a polyhedron with squares:
3 faces converge at each vertex
3 x 90°=270°<360° (hexahedron)
It is impossible for 4 or more faces to converge in one vertex because: 4 x 90°=360°
To construct a polyhedron with pentagons:
3 x 108°=324°<360° (dodecahedron)
It is impossible for 4 or more faces to converge in one vertex because: 4 x 108°=432°>360°

12. An outline of history of Polyhedra
Hexahedron - earth
Icosahedron - water
Octahedron - air
humid
cold
Tetrahedron - fire
hot
dry

13. Leonardo Pisano known as Fibonacci “Practica Geometriae”
(1222)
Piero della Francesca
“De quinque corporibus regularibus”
(Second half of the 15th century)
Luca Pacioli
“De Divina Proportione”
(1509)

14. “Mysterium Cosmographicum” 1596
Leonardo da Vinci
Johannes Kepler
“Mysterium Cosmographicum” 1596

15. Dual polyhedra Q P dual 12 vertices 20 vertices 30 edges 30 edges
12 faces
12 vertices
30 edges
20 faces

16. Dual polyhedra Q P dual 6 vertices 12 edges 8 faces 8 vertices

17. Process to convert a polyhedron P to its dual Q
Consider as vertices of Q the centres of the faces of P
The edges of Q are the segments that connect the centres of sequential faces of P
The faces of Q are the polygons that have as vertices the centre of the faces of P P Q P P

18. and side faces that are parallelograms
The prism
A prism is a polyhedron bounded by two bases, that are congruent polygons placed on parallel planes,
and side faces that are parallelograms
The distance between the planes containing the bases is the height of the prism
base
height
Side face
base

19. A prism with six rectangular faces is called rectangular prism
If the side faces are perpendicular to the planes of the bases, the prism is called a right prism; and, if the bases are regular polygons, the prism is called a regular prism
A prism with six rectangular faces is called rectangular prism
A prism with six faces made by parallelograms is called parallelepiped
rectangular prism
regular prism
parallelepiped

20. The pyramid
Consider a solid angle with vertex “V” and a plane “α” not passing through “V”
The part of solid angle containing “V” and delimited by “α” is called pyramid
V vertex
ABCDEF base
VAB lateral face (triangle)
VH height (distance vertex V and plane α)
VB lateral edge
AB edge base V α D E C H A B

21. The apothem (VM) of a right pyramid is the height of one of its faces
A pyramid is right if its base polygon circumscribes a circle and the base point of the pyramid height corresponds to the centre of the circle
The apothem (VM) of a right pyramid is the height of one of its faces
A pyramid is called regular if it is right and the base polygon is a regular polygon V α V
regular pyramid α C M
right pyramid

22. Surface area calculus
The faces of a polyhedron are poligons and we can therefore imagine to open the polyhedron and extend the faces on a plane
The surface area of a polyhedron is equal to the sum of the area of all of its faces
The results of the plane figure that we obtain take the name of development plane of the polyhedron

23. Volume solids
Two solids having the same spatial extension or volume are called equivalent
If two solids can be divided in an equal number of congruent solids, then they are equivalent
This is a sufficient but not necessary condition for equivalence between solids

24. Cavalieri's Principle
If parallel planes intersect two solids so that each plane defines equivalent sections, then two solids are equivalent that is the volumes of the two solids are equal
This is a sufficient but not necessary condition for equivalence between solids . P P’ α S S’ α’

25. Therefore the volume of a pyramid is
For this reason two prisms having equivalent bases and congruent height have equal volume:
Vprism =Sb h
Two pyramids with equivalent bases and congruent heights have equal volume
It is possible to demonstrate that the pyramid’s volume corresponds to a third of the volume of a prism with base and height congruent to those of the pyramid
Therefore the volume of a pyramid is
V pyramid =1/3 Sb h

26. Special thanks to prof. Cinzia Cetraro for linguistic supervision
Some of the pictures are taken from Wikipedia