1. 11.4 Exploring Solids
Geometry
How many geometric solid can you name?

2. Geometry 11.4 Exploring Solids
Goals
Know terminology about solids.
Identify solids by type.
Use Euler’s Theorem to solve problems.
May 12, 2019
Geometry 11.4 Exploring Solids

3. Geometry 11.4 Exploring Solids
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
Face
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Geometry 11.4 Exploring Solids

4. Geometry 11.4 Exploring Solids
Polyhedron
Many
Sides
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Geometry 11.4 Exploring Solids

5. Geometry 11.4 Exploring Solids
Polyhedron Views
Solid
Wire Frame
All three views will be used in these presentations, the text and other materials.
Hidden Line
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Geometry 11.4 Exploring Solids

6. Which of these are Polyhedrons?NO
YES
YES NO
YES
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Geometry 11.4 Exploring Solids

7. Geometry 11.4 Exploring Solids
Concave Polyhedra
A diagonal, or part of a diagonal, is outside the figure.
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Geometry 11.4 Exploring Solids

8. Geometry 11.4 Exploring Solids
Regular Polyhedra
All of the faces are congruent, regular polygons.
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Geometry 11.4 Exploring Solids

9. Geometry 11.4 Exploring Solids
Cross Section
The intersection of a solid and a plane.
Cross section is a circle.
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Geometry 11.4 Exploring Solids

10. Geometry 11.4 Exploring Solids
Cross Section
What is the intersection now?
Cross section is a rectangle.
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Geometry 11.4 Exploring Solids

11. What would the cross section be?
A Square
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Geometry 11.4 Exploring Solids

12. Geometry 11.4 Exploring Solids
Leonard Euler
1707 – 1783
One of the greatest mathematician of all time.
Worked in, and made enormous contributions to, every branch of mathematics.
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Geometry 11.4 Exploring Solids

13. Geometry 11.4 Exploring Solids
Euler’s Formula
Count F, the number of faces. 4 3 1 5 6
F = 6 2
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Geometry 11.4 Exploring Solids

14. Geometry 11.4 Exploring Solids
Euler’s Formula 7 6 5 8 2 3
Count V, the number of vertices. 1 4
V = 8
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Geometry 11.4 Exploring Solids

15. Geometry 11.4 Exploring Solids
Euler’s Formula 7 6 10 5 8 11 3 12 9 2 1 4
Count E, the number of edges.
E = 12
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Geometry 11.4 Exploring Solids

16. Geometry 11.4 Exploring Solids
Euler’s Formula
Faces =6
Vertices = 8
Edges = 12
V + F = E + 2
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Geometry 11.4 Exploring Solids

17. Geometry 11.4 Exploring Solids
Euler’s Formula
Faces = 6
Vertices = 8
Edges = 12
6 + 8 =
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Geometry 11.4 Exploring Solids

18. Geometry 11.4 Exploring Solids
Euler’s Formula
Faces = 6
Vertices = 8
Edges = 12
6 + 8 =
14 = 14
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Geometry 11.4 Exploring Solids

19. Geometry 11.4 Exploring Solids
Euler’s Formula
V + F = E + 2
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Geometry 11.4 Exploring Solids

20. Geometry 11.4 Exploring Solids
Try another figure…
Faces =
Vertices =
Edges =
F + V = E + 2
5 + 5 = 8 + 2
10 = 10
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Geometry 11.4 Exploring Solids

21. Geometry 11.4 Exploring Solids
Euler’s Formula
V + F = E + 2
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Geometry 11.4 Exploring Solids

22. Geometry 11.4 Exploring Solids
Solve:
A polyhedron has 8 faces and 12 vertices. How many edges does it have? 18
V + F = E + 2
= E + 2
20 = E + 2
E = 18
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Geometry 11.4 Exploring Solids

23. Geometry 11.4 Exploring Solids
Solve:
A polyhedron has 24 vertices and 36 edges. How many faces does it have? 14
V + F = E + 2
24 + F =
24 + F = 38
F = 14
May 12, 2019
Geometry 11.4 Exploring Solids

24. Geometry 11.4 Exploring Solids
Solve:
A polyhedron has 32 faces and 60 edges. How many vertices does it have? 30
V + F = E + 2
V + 32 =
V + 32 = 62
V= 30
May 12, 2019
Geometry 11.4 Exploring Solids

25. Geometry 11.4 Exploring Solids
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.
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Geometry 11.4 Exploring Solids

26. The Five Platonic Solids
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Geometry 11.4 Exploring Solids

27. Tetrahedron Has four triangular sides. May 12, 2019
Geometry 11.4 Exploring Solids

28. Geometry 11.4 Exploring Solids
Hexahedron (cube)
Has six square sides.
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Geometry 11.4 Exploring Solids

29. Octahedron Has eight triangular sides. May 12, 2019
Geometry 11.4 Exploring Solids

30. Geometry 11.4 Exploring Solids
Dodecahedron
Has 12 pentagonal faces.
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Geometry 11.4 Exploring Solids

31. Geometry 11.4 Exploring Solids
Icosahedron
Has 20 triangular faces.
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Geometry 11.4 Exploring Solids

32. Public Toilets in South Korea
This is not a Platonic Solid. It is a compound
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Geometry 11.4 Exploring Solids

33. Geometry 11.4 Exploring Solids
Summary
A polyhedron is a solid object.
The sides are faces.
Regular polyhedra have congruent faces.
There are 5 regular polyhedra (the Platonic Solids).
Euler’s Formula: F + V = E + 2
May 12, 2019
Geometry 11.4 Exploring Solids