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GEOMETRY – SOLID SHAPES
Up to now in geometry we have worked with flat shapes on a flat surface (such as
a piece of paper). Of course many things in our world are not flat, they are solid.
But some of the things which are true about flat shapes can also be applied to
solid shapes, if we think carefully about what is happening.
One nice thing about flat shapes is that they can be shown correctly on flat paper.
That does not work with solid shapes. If we try to show a solid shape on flat
paper, we must combine a true picture of a flat shape with an artistic
representation of the solid shape we really want to work with.
For example, look at a very simple solid shape, the cube. It is constructed from a
series of squares. But when we draw a picture of it, the lines must be shifted to
give the correct visual impression.
Sample A: picture of a cube:
In this picture of a cube, the front surface is a true square. The top and the left
side should also be true squares, but to give a convincing impression of a solid
figure they have been twisted out of their true shape in the picture. The dotted
lines stand for other lines which we know must exist at the back of the shape,
even though they can not actually be seen from in front where we are.
The reality of a true cube is six perfect squares assembled together in one solid
shape. The cube has eight corners. Every single angle of the cube is a right angle.
Every edge of the cube is in length equal to all the others. And the edges are
made with three sets of parallel lines. Each set has four edges, that is, four lines,
all parallel to each other.
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With all solid shapes, we must remember that the picture we use is just a visual
representation. Because the picture is not accurate in all its details, we must
know independently of their pictures the definitions of the shapes we are
dealing with.
A rectangular solid is much like a cube, but the edges are not all the same length.
Within each set of four parallel edges, the lines are all of equal length. But the
three sets of parallel lines may come in two or even three different lengths, one
for each set.
Sample B: picture of a rectangular solid with edges of two different lengths:
Sample C: picture of a rectangular solid with edges of three different lengths:
Like a cube, the rectangular solid has only right angles. The sides may be squares
or rectangles. (If all the sides are squares, it is a cube.) The pairs of opposite
sides—front/back; left/right; top/bottom—are always perfectly matched square
or oblong rectangles. That means for each side (or you could call it a surface)
there are two pairs of parallel lines of equal length, and all the angles are right.
These rectangular solids are examples from a larger category called prisms. That
definition calls for a pair of matching ends which are polygons such as triangle,
rectangle, pentagon, and so on. In some, those two ends are lined up at right
angles, and they are joined by rectangles. Several are shown in Sample D.
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Sample D: Assorted right regular prisms with rectangular sides:
(It is also possible to have right prisms which are not regular. This means that
their ends, although matching, are irregular rather than regular polygons.)
(In other prisms the two matching ends are parallel to each other, but offset.
That means the ends are not lined up with each other by right angles. Because of
that, the other sides are parallelograms, not rectangles. These are called oblique
prisms.)
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Sample E: Assorted regular prisms.
The prism in the center is oblique. Its four sides (but not the two ends) are
parallelograms
Another category of solid shapes is the pyramid. The right regular pyramid is a
solid figure with a regular polygon for a base. It has an isosceles triangle rising
from each edge of that base and meeting at a single point (the vertex) directly
above the center of the base. Since the base is a regular polygon, all its sides are
equal. Also, since the distance from any corner of the base to the vertex is always
the same, each of those sides is an identical isosceles triangle.
Sample F: Triangular pyramid (right and regular):
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Sample G: Square pyramid (right and regular):
Sample H: Pentagonal pyramid (right and regular):
(Just as prisms can be oblique, so can pyramids. That means the vertex is offset at
an oblique angle to the base, not lined up at a right angle.)
(It is also possible for the base of a pyramid to be an irregular polygon rather than
a regular polygon.)
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There are also solid shapes that incorporate curved lines such as circles. A right
regular cylinder is a solid shape somewhat like a prism. Its ends are matching
circles. Because circles have no flat edges, the connections from one end to the
other can not be parallelograms. But the connections act like a huge number of
parallel lines made from a rectangle rolled around the two circular ends. Viewed
from any side, it has a silhouette of a rectangle.
Sample I: Cylinder (right circular):
A right regular cone is a solid shape somewhat like a cylinder since it has a circle
at one end. A cone is also somewhat like a pyramid since from that circular base
its surface rises to a vertex directly over the center of the circular base. Viewed
from any point around the base, the cone has a silhouette of an isosceles triangle.
Sample J: Cone (right circular):
(The cylinder and the cone can also be oblique rather than right. And they can be
irregular, if the base is not a simple circle.)
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A sphere is a solid shaped like a round ball, such as a baseball or a basketball. It
has a curved surface, but no straight lines or angles. Every point on the surface of
a sphere is an equal distance from its center. Viewed from any direction, the
sphere has a silhouette of a circle.
Sample K : Sphere:
(If the sphere is squished or stretched, it is called a spheroid.)
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To help you answer the following questions, you may look back at the previous
pages.
CUBE has
all its sides in the shape of ____________________
what kind of angles? _______________
how many corners? ______________
how many edges? ______________
how many sets of parallel lines? _______________
how many parallel lines in each set? _______________
RECTANGULAR SOLID (excluding the cube) has
all or some of its sides in the shape of ____________________
what kind of angles? _______________
how many corners? ______________
how many edges? ______________
how many sets of parallel lines? _______________
how many parallel lines in each set? _______________
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RIGHT REGULAR PRISM (excluding the rectangular solid and the cube) has
two ends which are at ___________ angles to each other
two ends which are ________________ to each other
two ends which are ________________ to each other
two ends which are in the shape of ____________ _____________
sides which are in the shape of __________________
as many sides as there are __________ at either end
OBLIQUE REGULAR PRISM has
two ends which are at ___________ angles to each other
two ends which are ________________ to each other
two ends which are ________________ to each other
two ends which are in the shape of ____________ _____________
sides which are in the shape of __________________
as many sides as there are __________ at either end
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RIGHT REGULAR PYRAMID has
at its base a _______________ _____________
at the opposite end a vertex at ____________ angles to its base
as many sides as there are ___________ in its base
sides which are in the shape of ______________ ________________
OBLIQUE REGULAR PYRAMID has
at its base a _______________ _____________
at the opposite end a vertex at ______________ angles to its base
as many sides as there are ___________ in its base
sides which are in the shape of ______________ ________________
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RIGHT REGULAR CYLINDER has
from the front a silhouette of a ________________
two ends which are at ___________ angles to each other
two ends which are ________________ to each other
two ends which are ________________ to each other
two ends which are in the shape of ____________ _____________
RIGHT REGULAR CONE has
from the front a silhouette of an ________________ _______________
at its base a _______________
at the opposite end a vertex at an ______________ angle to its base
SPHERE has
from the front a silhouette of a ________________
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