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

2. Geometry 11.4 Three-Dimensional Figures
Warmup
May 12, 2019
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
May 12, 2019
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
May 12, 2019
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
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Geometry 11.4 Three-Dimensional Figures

6. Geometry 11.4 Three-Dimensional Figures
Polyhedron
Many
Sides
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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
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Geometry 11.4 Three-Dimensional Figures

8. Geometry 11.4 Three-Dimensional Figures
Types of Solids
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Geometry 11.4 Three-Dimensional Figures

9. 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.
May 12, 2019
Geometry 11.4 Three-Dimensional Figures

10. Name each solid. Which of these are Polyhedrons?
Pentagonal Prism
YES
Cylinder NO
Triangular Prism
YES
Square Pyramid
YES
Sphere NO
May 12, 2019
Geometry 11.4 Three-Dimensional Figures

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

12. Geometry 11.4 Three-Dimensional Figures
Regular Polyhedra
All of the faces are congruent, regular polygons.
May 12, 2019
Geometry 11.4 Three-Dimensional Figures

13. 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.
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Geometry 11.4 Three-Dimensional Figures

14. 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.
May 12, 2019
Geometry 11.4 Three-Dimensional Figures

15. 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

16. 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

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

18. 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.
May 12, 2019
Geometry 11.4 Three-Dimensional Figures

19. The Five Platonic Solids
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Geometry 11.4 Three-Dimensional Figures

20. Tetrahedron Has four triangular sides. Associated with fire.
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Geometry 11.4 Three-Dimensional Figures

21. Geometry 11.4 Three-Dimensional Figures
Hexahedron (cube)
Has six square sides.
Associated with earth.
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Geometry 11.4 Three-Dimensional Figures

22. Octahedron Has eight triangular sides. Associated with air.
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Geometry 11.4 Three-Dimensional Figures

23. Dodecahedron Has 12 pentagonal faces. Associated with the heavens.
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Geometry 11.4 Three-Dimensional Figures

24. Geometry 11.4 Three-Dimensional Figures
Icosahedron
Has 20 triangular faces.
Associated with water.
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Geometry 11.4 Three-Dimensional Figures

25. 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.
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Geometry 11.4 Three-Dimensional Figures

26. Geometry 11.4 Three-Dimensional Figures
Dungeons and Dragons
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Geometry 11.4 Three-Dimensional Figures

27. Public Toilets in South Korea
This is not a Platonic Solid. It is a compound polyhedron. Can you find out its correct name?
May 12, 2019
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.
May 12, 2019
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.
May 12, 2019
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.
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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.
May 12, 2019
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.
May 12, 2019
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:
May 12, 2019
Geometry 11.4 Three-Dimensional Figures