12.1 3-Dimensional Figures

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12.1 3-Dimensional Figures
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Objectives

• Use orthogonal drawings of 3-dimensional figures
  to make models
• Identify and use 3-dimensional figures

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Orthogonal Drawings

• To illustrate a 3-dimensional figure with
  2-dimensional views we use a method called
  orthogonal drawings.
• Orthogonal drawings are composite views of the
  top, left, front, and right sides of an object.
  Use the applets below to get a better
  understanding.
• 3-D Object Viewer
• Guess the View
• Coloring 3-D Sides
• Coloring 2-D Sides
• Rotating Blocks

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Example 1a
Draw the back view of a figure given its
orthogonal drawing.
If possible, use blocks to make a model. Then use
your model to draw the back view. Recognize that
bolder lines represent breaks in the surface.
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Example 1a
The top view indicates 1 row of different heights
and 1 column in the front right.
The front view indicates that there are four
standing columns. The first column to the left is
2 blocks high, the second column is 3 blocks
high, the third column is 2 blocks high, and the
fourth column to the far right is 1 block high.
The bolder segments indicate breaks in the
surface.
The right view indicates that the front right
column is only 1 block high. The bolder segments
indicate breaks in the surface.
Check the left side of your model. The back left
column is 2 blocks high. The bolder segments
indicate breaks in the surface.
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Example 1a
Check to see that all views correspond to the
model.
Now that your model is accurate, turn it around
to the back and draw what you see. The blocks are
flush, so no heavy segments are needed.
Answer
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Example 1b
Draw the corner view of the figure.
Turn your model so you are looking at the corners
of the blocks. The lowest columns should be in
the front so the differences in height between
the columns are visible.
Connect the dots on isometric dot paper to
represent the edges of the solid. Shade the tops
of each column.
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Example 1b
Answer
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Your Turn
a. Given its orthogonal drawing draw the back
view of the figure.
Answer
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Properties of Polyhedra

• A polyhedron is a solid bounded by polygons
  called faces which enclose a single region of
  space.
• An edge of a polyhedron is a line segment
  formed by the intersection of two faces.
• A vertex of a polyhedron is a point where three
  or more edges meet.
• The plural of polyhedron is either polyhedra or
  polyhedrons.

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Identifying Polyhedra

• Decide whether each of the solids below is a
  polyhedron. If so, count the number of faces,
  vertices, and edges of the polyhedron.

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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.

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Types of Solids
The prism has two parallel congruent faces called
bases. The other faces are parallelograms the
pyramid has all but one of its faces (the base)
intersecting at one vertex Both solids are named
by the shape of their bases followed by prism
or pyramid as their surname (i.e. triangular
prism and square pyramid).
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Regular Prisms and Regular Polyhedra

• If the bases of a prism are regular polygons,
  then it is called a regular prism.
• If all of the faces of a polyhedron are regular
  congruent polygons and all of the edges are
  congruent then it is called a regular polyhedron.
  

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Platonic Solids

• There are five (5) regular polyhedra called
  Platonic Solids named after the Greek
  mathematician and philosopher, Plato.
• The five Platonic Solids are a regular
  tetrahedron (see above),

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Platonic Solids

• a cube (6 faces)

a regular dodecahedron

• a regular octahedron (8 faces),

and an icosahedron .
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Example 2a
Identify the solid. Name the bases, faces, edges,
and vertices.
The bases are rectangles and the four remaining
faces are parallelograms.
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Example 2a
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Example 2b
Identify the solid. Name the bases, faces, edges,
and vertices.
The bases are circular and congruent.
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Example 2c
Identify the solid. Name the bases, faces, edges,
and vertices.
The base is a triangle, and the remaining three
faces meet at a point.
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Your Turn
a. Identify the solid. Name the bases, faces,
edges, and vertices.
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Your Turn
b. Identify the solid. Name the bases, faces,
edges, and vertices.
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Your Turn
c. Identify the solid.
Answer pentagonal pyramid
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Cross Sections

• Imagine a plane slicing through a solid. The
  intersection of the plane and the solid is called
  a cross section. For instance, the diagram above
  shows that the intersection of a plane and a
  sphere is a circle.

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Describing Cross Sections

• Describe the shape formed by the intersection of
  the plane and the cube.

This cross section is a square.
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Describing Cross Sections

• Describe the shape formed by the intersection of
  the plane and the cube.

This cross section is a pentagon.
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Describing Cross Sections

• Describe the shape formed by the intersection of
  the plane and the cube.

This cross section is a triangle.
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Example 3
A customer orders a two-layer sheet cake.
Describe the possible cross sections of the cake.
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Example 3
Answer
If the cake is cut horizontally, the cross
section will be a rectangle.
If the cake is cut vertically, the cross section
will also be a rectangle.
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Example 3
A solid cone is going to be sliced so that the
resulting flat portion can be dipped in paint and
used to make prints of different shapes. How
should the cone be sliced to make prints of a
circle, triangle, and an oval?
Answer If the cone were to be cut parallel to
the base, the cross-section would be a
circle.
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Example 3
AnswerIf the cone were to be cut perpendicular
to the base, the slice would be a triangle.
If the cone were to be cut on an angle to the
base, the slice would be an oval.
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Assignment

• Pg. 640
• 9, 11, 13, 15, 16 21, 25 - 30