Polygons

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Polygons

• A many sided figure

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The cross section of a brilliant-cut diamond
forms a pentagon. The most beautiful and
valuable diamonds have precisely cut angles that
maximize the amount of light they reflect.A
pentagon is a type of polygon.Prefixes are used
to name different types of polygons.
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Polygon a closed plane figure formed by three
or more segments.Regular polygon a polygon
with congruent sides and angles.Prefixes used
to name polygons tri-, quad-, penta-, hexa-,
hepta-, octa-, nona-, deca-Polygons are named
(classified) based on the number of sides.
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PolygonsProperties of polygons, interior angles
of polygons including triangles, quadrilaterals,
pentagons, heptagons, octagons, nonagons, and
decagons.

• Properties of Triangles
• Triangle 3-sided polygon
• The sum of the angles in any triangle is 180
  (triangle sum theorem)

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The formula we use to find the sum of the
interior angles of any polygon comes from the
number of triangles in a figure
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First remember that the sum of the interior
angles of a polygon is given by the formula
180(n-2).A polygon is called a REGULAR when all
the sides are congruent and all the angles are
congruent.The picture shown to the left is that
of a Regular Pentagon. We know that to find the
sum of its interior angles we substitute n 5 in
the formula and get180(5 -2) 180(3) 540
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Regular triangles - EquilateralAll sides are the
same length (congruent) and all interior angles
are the same size (congruent).To find the
measure of the interior angles, we know that the
sum of all the angles equal 180, and there are
three angles.So, the measure of the interior
angles of an equilateral triangle is 60.
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Quadrilaterals squaresAll sides are the same
length (congruent) and all interior angles are
the same size (congruent)To find the measure of
the interior angles, we know that the sum of the
angles equal 360, and there are four angles, so
the measure of the interior angles are 90.
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Pentagon a 5-sided polygonTo find the sum of
the interior angles of a pentagon, we divide the
pentagon into triangles. There are three
triangles and because the sum of each triangle is
180 we get 540, so the measure of the interior
angles of a regular pentagon is 540
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Hexagon a 6-sided polygonTo find the sum of
the interior angles of a hexagon we divide the
hexagon into triangles. There are four triangles
and because the sum of the angles in a triangle
is 180, we get 720, so the measure of the
interior angles of a regular hexagon is 720.
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Octagon an 8-sided polygonAll sides are the
same length (congruent) and all interior angles
are the same size (congruent).What is the sum of
the angles in a regular octagon?
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Nonagon a 9-sided polygonAll sides are the
same length (congruent) and all interior angles
are the same size (congruent).What is the sum
of the interior angles of a regular nonagon?
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Decagon a 10-sided polygonAll sides are the
same length (congruent) and all interior angles
are the same size (congruent).What is the sum
of the interior angles of a regular decagon?
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Using the pentagon example, we can come up with a
formula that works for all polygons.Notice that
a pentagon has 5 sides, and that you can form 3
triangles by connecting the vertices. Thats 2
less than the number of sides. If we represent
the number of sides of a polygon as n, then the
number of triangles you can form is (n-2). Since
each triangle contains 180, that gives us the
formulasum of interior angles 180(n-2)
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Warning !

• Look at the pentagon to the right. Do angle E
  and angle B look like they have the same
  measures? Youre right---they dont. This
  pentagon is not a regular pentagon.
• If the angles of a polygon do not all have the
  same measure, then we cant find the measure of
  any one of the angles just by knowing their sum.

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Using the Formula

• Example 1 Find the number of degrees in the sum
  of the interior angles of an octagon.
• An octagon has 8 sides. So n 8. Using our
  formula, that gives us 180(8-2) 180(6) 1080

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Example 2 How many sides does a polygon have if
the sum of its interior angles is 720?Since,
this time, we know the number of degrees, we set
the formula equal to 720, and solve for
n.180(n-2) 720 set the formula 720n - 2
4 divide both sides by 180 n 6
add 2 to both sides
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Names of Polygons

• Triangle 3 sides
• Quadrilateral 4 sides
• Pentagon 5 sides
• Hexagon 6 sides
• Heptagon or Septagon 7 sides
• Octagon 8 sides
• Nonagon or Novagon 9 sides
• Decagon 10 sides

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Practice with Sum of Interior Angles

• The sum of the interior angles of a hexagon.
• 360
• 540
• 720

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How many degrees are there in the sum of the
interior angles of a 9-sided polygon?a)
1080b) 1260c) 1620
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If the sum of the interior angles of a polygon
equals 900, how many sides does the polygon
have?a) 7b) 9c) 10
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How many sides does a polygon have if the sum of
its interior angles is 2160?a) 14b) 16c)
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What is the name of a polygon if the sum of its
interior angles equals 1440?a) octagonb)
decagonc) pentagon
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Special Quadrilaterals

• 4-sided figures

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Quadrilaterals with certain properties are given
additional names.
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A square has 4 congruent sides and 4 right angles.
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A rectangle has 4 right angles.
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A parallelogram has 2 pairs of parallel sides.
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A rhombus has 4 congruent sides.
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A kite has 2 sets of adjacent sides that are the
same length (congruent) and one set of opposite
angles that are congruent.