Know terminology about solids. Identify solids by type. Use Euler’s Theorem to solve problems.
3April 19, 2023
Polyhedron
A solid that is bounded by polygons.
The polygons are faces.
An edge is the intersection of two faces.
A vertex is the intersection of three or more faces.
Face
Face
Fac
e
4April 19, 2023
Polyhedron
5April 19, 2023
Polyhedron Views
SolidWire Frame
Hidden Line
All three views will be used in these presentations, the text and other materials.
6April 19, 2023
Which of these are Polyhedrons?
YES
YESNO
NO YES
7April 19, 2023
Concave PolyhedraA diagonal, or part of a diagonal, is outside the figure.
8April 19, 2023
Regular Polyhedra
All of the faces are congruent, regular polygons.
9April 19, 2023
Cross Section
The intersection of a solid and a plane.
Cross section is a circle.
10April 19, 2023
Cross Section
What is the intersection now?
Cross section is a rectangle.
11April 19, 2023
What would the cross section be?
A Square
12April 19, 2023
Leonard Euler
1707 – 1783 Probably the greatest
mathematician of all time.
Worked in, and made enormous contributions to, every branch of mathematics.
13April 19, 2023
Euler’s Formula
Count F, the number of faces.
123456 F = 6
14April 19, 2023
Euler’s Formula
Count V, the number of vertices.
1
23
4
56
V = 8
78
15April 19, 2023
Euler’s Formula
Count E, the number of edges.
12
34
56
E = 12
78
9
1011
12
16April 19, 2023
Euler’s Formula
Faces =
6
Vertices =
8
Edges =
12
V + F = E + 2
17April 19, 2023
Euler’s Formula
Faces =
6
Vertices =
8
Edges =
12
6 + 8 = 12 + 2
18April 19, 2023
Euler’s Formula
Faces =
6
Vertices =
8
Edges =
12
6 + 8 = 12 + 2
14 = 14
19April 19, 2023
Euler’s Formula
20April 19, 2023
Try another figure…Faces =
Vertices =
Edges =
F + V = E + 2
5 + 5 = 8 + 2
10 = 10
21April 19, 2023
Euler’s Formula
22April 19, 2023
Solve:
A polyhedron has 8 faces and 12 vertices. How many edges does it have?
18 V + F = E + 2 12 + 8 = E + 2 20 = E + 2 E = 18
23April 19, 2023
Solve:
A polyhedron has 24 vertices and 36 edges. How many faces does it have?
14 V + F = E + 2 24 + F = 36 + 2 24 + F = 38 F = 14
24April 19, 2023
Solve:
A polyhedron has 32 faces and 60 edges. How many vertices does it have?
30 V + F = E + 2 V + 32 = 60 + 2 V + 32 = 62 V= 30
25April 19, 2023
The Platonic Solids
There are only five of them. They are regular, convex polyhedra. First described ca. 350 BC by Plato in
Timaeus. Have been found in many ancient cultures.
26April 19, 2023
The Five Platonic Solids
27April 19, 2023
Tetrahedron
Has four triangular sides.
Associated with fire.
28April 19, 2023
Hexahedron (cube)
Has six square sides.
Associated with earth.
29April 19, 2023
Octahedron
Has eight triangular sides.
Associated with air.
30April 19, 2023
Dodecahedron
Has 12 pentagonal faces.
Associated with the heavens.
31April 19, 2023
Icosahedron
Has 20 triangular faces.
Associated with water.
32April 19, 2023
Johannes Kepler In 1596 Kepler published a
tract called The Cosmic Mystery in which he envisioned the universe as consisting of nested Platonic Solids whose inscribed spheres determine the orbits of the planets, all enclosed in a sphere representing the outer heavens.
33April 19, 2023
Dungeons and Dragons
34April 19, 2023
Public Toilets in South KoreaThis is not a Platonic Solid. It is a compound polyhedron. Can you find out its correct name?
35April 19, 2023
Platonic Solid Links
Mathworld
GSP Icosahedron
36April 19, 2023
Summary
A polyhedron is a solid object. The sides are faces. Regular polyhedra have congruent faces. There are 5 regular polyhedra (the