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12.1 Exploring Solids

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Title: 12.1 Exploring Solids Author: Robert Spitz Last modified by: Robert Spitz Created Date: 5/8/2005 10:52:12 PM Document presentation format: On-screen Show – PowerPoint PPT presentation

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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!!!
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