Transcript LESSON THREE: IF I HATE THIS MOVIE…

LESSON THIRTY-FIVE:
ANOTHER DIMENSION
THREE-DIMENSIONAL FIGURES
• As you have certainly realized by now, objects
in the real world do not exist in a two
dimensional plane.
• The real world in its entirety exists in three
dimensions.
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• You may remember from algebra and a bit
from this class also that when we graph
points, we generally do it on an x-y plane,
hence the name “two dimensional”.
• we add a new dimension when dealing with
three dimensional figures, logically named
the z-plane.
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• Working with three dimensional figures is
where the money is!
• Engineers, CAD Artists, Advertising
Executives, Architects and especially
Videogame Programmers, are constantly
using 3D models in their work.
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• As all these professions know, there are many
different perspectives to any 3D figure.
• You can think of a perspective as the angle
from which you view an object.
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• There are two main views we can examine of
three-dimensional figures.
• The first is front view.
• This is where the front or face of an object is
considered centered.
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• The second is isometric view.
• This is where the corner of an object is
considered centered. You will need to make
these on what is called isometric dot paper.
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• You may be asked to draw a rectangular prism
with a length of 3, width of 4 and height of 5.
• The following slide shows this on isometric
dot paper.
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• We have two different definitions for the
“sides” of a 3-dimensional figure.
• We call the bottom and top surfaces of a 3dimensional figure, the bases.
• We call any flat surface of a 3-dimensional
figure a face.
• A base is just a special type of face.
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• With that, we must discuss the various types
of three dimensional figures.
• The first and simplest is a prism.
• Prisms are three-dimensional objects with
two congruent and parallel faces.
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• There are many types of prisms.
• The easiest is a rectangular prism.
• In these…
– There are six faces.
– All faces meet at 90.
– Opposite faces are parallel.
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• A cube is a special type of this in which all
sides are equal in area and are squares.
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• The next is a triangular prisms.
• Simply enough, this is a prism in which the
bases are both triangles.
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• Beyond that, lie the polygonal prisms.
• These are prisms with regular or irregular
polygons for their bases.
• We name each of these by the figure that
makes its base.
– For example we would call the prism below
hexagonal and pentagonal respectively.
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• Prisms can be what we call right or oblique.
• A right prism is one in which the base edges
and the lateral edges all form right angles.
• An oblique prism is one in which not all the
base edges and lateral edges form right
angles.
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• After the prisms, come the pyramids.
• Pyramids are three-dimensional objects with
a polygon for the base, and triangles for faces.
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• A triangular pyramid is a pyramid whose base
is a triangle.
• A triangular pyramid that’s made up of four
equilateral triangles is called a tetrahedron.
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• Obviously, a square pyramid is a pyramid with
a square for its base.
• And a rectangular pyramid is one with a
rectangle for it’s base.
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• Polygonal pyramids are ones with polygons
for their bases.
• We will be dealing primarily pyramids that
have regular polygons for their bases.
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• Much like prisms, pyramids can be slanted or
what we would call “straight”.
• These are called simply regular and nonregular respectively.
• We’ll come back to this in a later lesson.
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• Next are the cylinders.
• Cylinders are three-dimensional objects with
two parallel, circular bases.
• Since these can only have a circle for the base,
these don’t really have “types” like our
previous figures.
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• The same can be said four our next figure, the
cones.
• Cones are a three-dimensional figure with a
circular base and a curved face that comes to
a point.
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• Finally, one of the most common real-world
objects and yet one of the most difficult is the
sphere.
• Our only definition of this is that it is a threedimensional figure in which all points are
equidistant from a center point.
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• In this unit, we will be learning about all the
figures just mentioned.
• Today, we will be focusing on prisms.
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• Prisms have many measures and many which
you will have worked with before.
• First is the surface area.
• This is the sum area of all the bases and faces
of a prism.
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• Most of these will be rectangles.
• Some may be regular polygons so remember
that A= ½ nsa or A = ½ Pa.
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• Second is the lateral area.
• This is the area of all the lateral faces
• Basically, think of this as the surface area
minus the area of the bases.
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• So let’s try a couple!
• What is the surface area and lateral area of
the prism below?
30cm
10 cm
12 cm
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• For the surface area we simply add the area of
all the faces.
• 2(30 x 10) + 2(12 x 10) + 2(30 x 12) =1560 cm²
30 cm
10 cm
12 cm
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• For the lateral area we simply don’t count the
two bases.
• You can say the bases are the top and bottom.
• 2(30 x 10) + 2(12 x 10) + 2(30 x 12) = 960 cm²
30 cm
10 cm
12 cm
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• What about this one?
• Well the area of the regular hexagons is ½nsa,
in this case ½(6)(5)(5) which equals 75 cm²
5 cm
7 cm
5 cm
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• There are two of these so the sum area of the
bases is 150 cm².
• All the rectangles have a dimensions of 5 x 7
and there are six of them, so their total area is
210cm² (this is the lateral area!)
• So the surface area is 360 cm².
7 cm
5
5 cm
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• We could have also figured the lateral area by
taking the perimeter of the polygon an
multiplying by the altitude of the prism.
• 30 cm x 7 cm = 210 cm²
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• You’ll find that the volume of prisms is much
easier to find.
• You all have heard length x width x height.
• This works for rectangular prisms but what of
the others?
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• A more general formula is Ba or base x altitude.
• For this, we take the area of the base and
multiply by the altitude.
• This will work for all prisms.
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• Take this example.
• I find the area of the hexagon ½ (6)(5)(15)
which equals 225 ft²
• Then I multiply by the altitude to get 2250 ft³.
10 ft
15 ft
10 ft
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• You will be asked to solve for lateral area,
surface area and volume of prisms today.
• In the coming days and weeks, we will cover
each of these for cones, cylinders, pyramids
and spheres as well.