1. 11.4 Three-Dimensional Figures
Geometry
11.4 Three-Dimensional Figures

2. Geometry 11.4 Three-Dimensional Figures
Warmup
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

3. Geometry 11.4 Three-Dimensional Figures
Essential Question
What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

4. Geometry 11.4 Three-Dimensional Figures
Goals
Classify solids
Describe cross sections
Sketch and describe solids of revolution
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

5. Geometry 11.4 Three-Dimensional Figures
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
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

6. Geometry 11.4 Three-Dimensional Figures
Polyhedron
Many
Sides
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

7. Geometry 11.4 Three-Dimensional Figures
Polyhedron Views
Solid
Wire Frame
All three views will be used in these presentations, the text and other materials.
Hidden Line
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

8. Geometry 11.4 Three-Dimensional Figures
Concave Polyhedra
A diagonal, or part of a diagonal, is outside the figure.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

9. Geometry 11.4 Three-Dimensional Figures
Regular Polyhedra
All of the faces are congruent, regular polygons.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

10. Geometry 11.4 Three-Dimensional Figures
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.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

11. The Five Platonic Solids
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

12. Tetrahedron Has four triangular sides. Associated with fire.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

13. Geometry 11.4 Three-Dimensional Figures
Hexahedron (cube)
Has six square sides.
Associated with earth.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

14. Octahedron Has eight triangular sides. Associated with air.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

15. Dodecahedron Has 12 pentagonal faces. Associated with the heavens.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

16. Geometry 11.4 Three-Dimensional Figures
Icosahedron
Has 20 triangular faces.
Associated with water.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

17. Geometry 11.4 Three-Dimensional Figures
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.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

18. Geometry 11.4 Three-Dimensional Figures
Dungeons and Dragons
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

19. Public Toilets in South Korea
This is not a Platonic Solid. It is a compound polyhedron. Can you find out its correct name?
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

20. Geometry 11.4 Three-Dimensional Figures
Types of Solids
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

21. Geometry 11.4 Three-Dimensional Figures
Naming Polyhedra
To name a prism or a pyramid, use the shape of the base. The two bases of a prism are congruent polygons in parallel planes. The base of a pyramid is a polygon.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

22. Name each solid. Which of these are Polyhedrons?
Pentagonal Prism
YES
Cylinder NO
Triangular Prism
YES
Square Pyramid
YES
Sphere NO
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

23. Geometry 11.4 Three-Dimensional Figures
Cross Section
The intersection of a solid and a plane.
The cross section is parallel to the bases.
Cross section is a circle.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

24. Geometry 11.4 Three-Dimensional Figures
Cross Section
What is the intersection now?
Cross section is a rectangle.
The cross section is perpendicular to the bases.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

25. What would the cross section be?
The cross section is parallel to the base.
A Square
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Geometry 11.4 Three-Dimensional Figures

26. What would the cross section be?
The cross section is perpendicular to both bases.
A Pentagon
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Geometry 11.4 Three-Dimensional Figures

27. What would the cross section be?
The cross section is slanted and intersects both bases.
A Trapezoid
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

28. Geometry 11.4 Three-Dimensional Figures
A solid of revolution is a three-dimensional figure that is formed by rotating a two-dimensional shape around an axis.
If you rotate a rectangle around one of its sides, the path it makes through space is a cylinder.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

29. Solids of Revolution Example: Sketching a Solid of Revolution
Draw the solid of revolution formed by the shape rotated around the axis given. Describe the resulting shape.
A cylinder with the center removed.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

30. Geometry 11.4 Three-Dimensional Figures
Check it Out! Example 1
Draw the solid of revolution formed by the shape rotated around the axis given. Describe the resulting shape.
A donut shape, also known as a torus.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

31. Find the axis of symmetry.
Example 2: Sketching a Solid of Revolution
Draw a two-dimensional shape and axis of rotation that could form each figure.
Find the axis of symmetry.
The two-dimensional shape should match the outline of one side of the shape.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

32. Geometry 11.4 Three-Dimensional Figures
Check it Out! Example 2
Draw a two-dimensional shape and axis of rotation that could form the sports drink bottle.
Find the axis of symmetry.
The two-dimensional shape should match the outline of one side of the shape.
November 17, 2018
Geometry 11.4 Three-Dimensional Figures

33. Geometry 11.4 Three-Dimensional Figures
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:
November 17, 2018
Geometry 11.4 Three-Dimensional Figures