Convex Polygons

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Convex Polygons
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Polygon

• A polygon is a closed plane figure whose sides
  are line segments that intersect only at the
  endpoints.
• Which of the following are polygons?

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Convex and Concave Polygons
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Familiar Polygons
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Polygon Names
11 hendecagon 12 dodecagon 3
triangle 13 tridecagon 4 quadrilateral
14 tetradecagon 5 pentagon 15
pendedecagon 6 hexagon 16 hexdecagon 7
heptagon 17 heptdecagon 8 octagon 18
octdecagon 9 nonagon 19
enneadecagon 10 decagon 20 icosagon
After about 6 sides mathematicians usually refer
to these polygons as n-gons. So your 13 sided
polygon I would call a 13-gon
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Icosahedron
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A Girls Best Polygon
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A Girls Best Polygon
girdle diameter 100 table 53 - 57
crown 16.2 girdle 1 - 2.5 pavilion 43.1
crown 34.5 pavillion 40.75 cutlet 98.5
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Carat
A diamond's weight is measured in carats, the
easiest of the 4 C's to determine. Each carat is
divided into 100 parts called points. A half
carat diamond is equal to 50 points or 0.50
carats. Carat weight does not refer to the size
of a stone. A 1 carat ruby will not have the same
geometric measurements as a one carat diamond.
Ruby's weigh more than diamonds. Two diamonds of
equal size can differ in value due to its color,
clarity, and cut. The price per carat can
increase exponentially as diamonds increase in
size. Two diamonds, weighing 0.50 carats and the
other weighing 1.00 carat, with equal cut, color,
and clarity will have significant cost
differences.
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Cut

• Cut, the arrangement of a diamond's facets, is
  one of the most important of all characteristics,
  and among the hardest to judge. A diamond's shape
  is not to be confused with its cut. The
  proportions of a stone, symmetry, polish, and
  precision of faceting determine how much of the
  diamonds potential fire and beauty may be
  released, its perceived size and even, its
  apparent color.

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Cut
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Color
GIA's color grading scale D - F Colorless the
most valuable and desirable color grade because
the maximum amount of light passes through this
color range. G - J Near Colorless these stones
appear colorless to the untrained eye and can
appear colorless after being mounted (especially
in yellow gold). K - M Faint Yellow slightly
tinted with yellow color to the untrained eye,
but still considered desirable for affordable
pieces of jewelry. N - R Very Light Yellow
visible yellow tint to the untrained eye. S - Z
Light Yellow highly noticeable yellow tint to
the untrained eye. Each letter marks a different
color grade. Increased shadings of yellow reduce
the value of a diamond, they do not necessarily
reduce its beauty. Well cut diamonds can disguise
coloration

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Clarity
F Flawless contain no internal or external
inclusions under 10 times magnification IF
Internally Flawless contain insignificant
surface imperfections, but no internal inclusions
VVS1-VVS2 Very, Very Slightly Included
inclusions are extremely difficult to find under
10 times magnification VS1-VS2 Very Slightly
Included inclusions are difficult to see under
10 times magnification and appear 100 clean to
the naked eye SI1-SI2 Slightly Included
noticeable inclusions that are easily found under
a 10 times magnification I1-I2 Included
contain obvious inclusions under 10 times
magnification and can be seen with an unaided
eye I3 Included contain noticeable internal and
external inclusions which are easily seen by an
untrained eye. A diamonds clarity is determined
by the number, location, and intensity of
inclusions such as crystals, clouds, or feathers
it contains. Inclusions, naturally occurring
imperfections, can be viewed under a 10 times
magnification. The fewer the inclusions, the more
valuable the stone. A well cut diamond can
minimize the appearance of inclusions.
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Regular Pentagon
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6
120?
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60?
6
6
6
6
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Regular Pentagon
Regular Pentagon n 5 which means 5 congruent
sides and 5 congruent angles Number of diagonals
from each vertex n-3 5-3 2 Total number of
diagonals n(n-3)/2 5(5-3)/2 5 Number of
triangles n-2 5 2 3 ? of exterior angles
360? each exterior ? 360?/n 360?/5
72? Each interior angle 180? - 72? 108? Each
interior angle (n-2)180/n (5-2)180/5 3
180 / 5 108? ? of interior angles (n-2)180
(5-2)180 3 180 540? ? of interior angles
Number of triangles 180? 3 180? 540?
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Diagonals

• A diagonal of a polygon is a line segment that
  joins two nonconsecutive vertices.

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Theorem

• Theorem The total number of diagonals D in a
  polygon of n sides is given by the formula.
• Check this formula for several polygons.

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Sum of the Interior Angles of a Polygon

• Theorem The sum S of the measure of the interior
  angles of a polygon with n sides is given by
• Note that n gt2 for any polygon.

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Example

• Find the sum of the interior angle of a pentagon.

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Example
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Example
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Regular Polygons

• A regular polygon is a polygon that is both
  equilateral and equiangular.

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Corollary

• Corollary The measure I of each interior angle
  of a regular polygon of n sides is
• Pick your favorite regular polygon and use the
  formula.

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Example

• Find the measure of a interior angle of a
  hexagon. (n6)

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Example
m
QNL
120
Q
N
J
L
O
P
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Corollary

• Corollary The sum of the four interior angles
  of a quadrilateral is 360 degrees. (look at the
  two triangles to check)

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Corollary

• Corollary The sum of the measure of the
  exterior angles of a polygon, one at each vertex,
  is 360 degrees.

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Example
R
m
RQN
60
N
S
Q
W
m
SNL
60
m
TLK
60
L
J
m
UOM
60
m
VPJ
60
T
V
P
O
m
WJQ
60
U
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Corollary

• Corollary The measure E of each exterior angle
  of a regular polygon of n sides is

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Polygon Convex Closed plane figures whose
sides are line segments, which intersect only at
their endpoints
always
Quadrilateral A polygon with four sides.
No pair of opposite sides or ?.
Parallelogram A quadrilateral Opposite sides are
? and . Opposite ?s are ?. Consecutive ?s are
supplementary Diagonals bisect each other.
Trapezoid Bases arebut not ? , sides not If
isosceles sides ?, base ?s ?, and diagonals ?.
Midsegment is to each base, length½(b1b2).
Rhombus A parallelogram with sides ?. Diagonals
are -, bisect each other and bisect opposite ?s
.
Rectangle A parallelogram with 90 degree angles.
Diagonals are congruent and bisect each other.
Square Four congruent sides and 90 degree
angles. A rectangle with congruent sides. A
rhombus with 90 degree ?s .
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Homework Chapter 7

• sec page problems
• 7.3 309 1, 2, 3, 4, 6, 10, 12, 13
• 7.4 316 1, 2, 3, 4,
• R 320 9, 11, 20