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## 12.1 Exploring Solids

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Title: 12.1 Exploring Solids

1
12.1 Exploring Solids
• Geometry
• Mrs. Spitz
• Spring 2006

2
Objectives/Assignment
• Use properties of polyhedra.
• Use Eulers Theorem in real-life situations, such
as analyzing the molecular structure of salt.
• You can use properties of polyhedra to classify
various crystals.
• Assignment 12.1 Worksheet A

3
Using properties of polyhedra
• A polyhedron is a solid that is bounded by
polygons called faces, that enclose a since
region of space. An edge of a polyhedron is a
line segment formed by the intersection of two
faces.

4
Using properties of polyhedra
• A vertex of a polyhedron is a point where three
or more edges meet. The plural of polyhedron is
polyhedra or polyhedrons.

5
Ex. 1 Identifying Polyhedra
• Decide whether the solid is a polyhedron. If so,
count the number of faces, vertices, and edges of
the polyhedron.

6
1. This is a polyhedron. It has 5 faces, 6
vertices, and 9 edges.
2. This is not a polyhedron. Some of its faces are
not polygons.
3. This is a polyhedron. It has 7 faces, 7
vertices, and 12 edges.

7
Types of Solids
8
Regular/Convex/Concave
• A polyhedron is regular if all its faces are
congruent regular polygons. A polyhedron is
convex if any two points on its surface can be
connected by a segment that lies entirely inside
or on the polyhedron.

9
continued . . .
• If this segment goes outside the polyhedron, then
the polyhedron is said to be NON-CONVEX, OR
CONCAVE.

10
Ex. 2 Classifying Polyhedra
• Is the octahedron convex? Is it regular?

It is convex and regular.
11
Ex. 2 Classifying Polyhedra
• Is the octahedron convex? Is it regular?

It is convex, but non- regular.
12
Ex. 2 Classifying Polyhedra
• Is the octahedron convex? Is it regular?

It is non-convex and non- regular.
13
Note
• Imagine a plane slicing through a solid. The
intersection of the plane and the solid is called
a cross section. For instance, the diagram shows
that the intersection of a plane and a sphere is
a circle.

14
Ex. 3 Describing Cross Sections
• Describe the shape formed by the intersection of
the plane and the cube.

This cross section is a square.
15
Ex. 3 Describing Cross Sections
• Describe the shape formed by the intersection of
the plane and the cube.

This cross section is a pentagon.
16
Ex. 3 Describing Cross Sections
• Describe the shape formed by the intersection of
the plane and the cube.

This cross section is a triangle.
17
Note . . . other shapes
• The square, pentagon, and triangle cross sections
of a cube are described in Ex. 3. Some other
cross sections are the rectangle, trapezoid, and
hexagon.

18
• Polyhedron a three-dimensional solid made up of
plane faces. Polymany Hedronfaces
• Prism a polyhedron (geometric solid) with two
parallel, same-size bases joined by 3 or more
parallelogram-shaped sides.
• Tetrahedron polyhedron with four faces
(tetrafour, hedronface).

19
Using Eulers Theorem
• There are five (5) regular polyhedra called
Platonic Solids after the Greek mathematician and
philosopher Plato. The Platonic Solids are a
regular tetrahedra

20
Using Eulers Theorem
• A cube (6 faces)
• dodecahedron
• A regular octahedron (8 faces),
• icosahedron

21
Note . . .
• Notice that the sum of the number of faces and
vertices is two more than the number of edges in
the solids above. This result was proved by the
Swiss mathematician Leonhard Euler.

Leonard Euler 1707-1783
22
Eulers Theorem
• The number of faces (F), vertices (V), and edges
(E) of a polyhedron are related by the formula
• F V E 2

23
Ex. 4 Using Eulers Theorem
• The solid has 14 faces 8 triangles and 6
octagons. How many vertices does the solid have?

24
Ex. 4 Using Eulers Theorem
• On their own, 8 triangles and 6 octagons have
8(3) 6(8), or 72 edges. In the solid, each
side is shared by exactly two polygons. So the
number of edges is one half of 72, or 36. Use
Eulers Theorem to find the number of vertices.

25
Ex. 4 Using Eulers Theorem
F V E 2
Write Eulers Thm.
14 V 36 2
Substitute values.
14 V 38
Simplify.
V 24
Solve for V.
?The solid has 24 vertices.
26
Ex. 5 Finding the Number of Edges
• Chemistry. In molecules of sodium chloride
commonly known as table salt, chloride atoms are
arranged like the vertices of regular
octahedrons. In the crystal structure, the
molecules share edges. How many sodium chloride
molecules share the edges of one sodium chloride
molecule?

27
Ex. 5 Finding the Number of Edges
• To find the of molecules that share edges with
a given molecule, you need to know the of edges
of the molecule. You know that the molecules
are shaped like regular octahedrons. So they
each have 8 faces and 6 vertices. You can use
Eulers Theorem to find the number of edges as
shown on the next slide.

28
Ex. 5 Finding the Number of Edges
F V E 2
Write Eulers Thm.
8 6 E 2
Substitute values.
14 E 2
Simplify.
12 E
Solve for E.
?So, 12 other molecules share the edges of the
given molecule.
29
Ex. 6 Finding the of Vertices
• SPORTS. A soccer ball resembles a polyhedron
with 32 faces 20 are regular hexagons and 12 are
regular pentagons. How many vertices does this
polyhedron have?

30
Ex. 6 Finding the of Vertices
• Each of the 20 hexagons has 6 sides and each of
the 12 pentagons has 5 sides. Each edge of the
soccer ball is shared by two polygons. Thus the
total of edges is as follows.

E ½ (6 20 5 12)
Expression for of edges.
½ (180)
Simplify inside parentheses.
90
Multiply.
?Knowing the of edges, 90, and the of faces,
32, you can then apply Eulers Theorem to
determine the of vertices.
31
Apply Eulers Theorem
F V E 2
Write Eulers Thm.
32 V 90 2
Substitute values.
32 V 92
Simplify.
V 60
Solve for V.
?So, the polyhedron has 60 vertices.
32
Upcoming
• There is a quiz after 12.3. There are no other
quizzes or tests for Chapter 12
• Review for final exam.
• Final Exams Scheduled for Wednesday, May 24.
You must take and pass the final exam to pass the
course!
• Book return You will turn in books/CDs this
date. No book returned F for semester! Book
is 75 to replace.
• Absences More than 10 in a semester from
January 9 to May 26, and I will fail you.
Tardies count!!!