Formulas Involving Polygons Chapter 7 Section 3

1
Formulas Involving PolygonsChapter 7 Section 3

• By Alex Pipcho

2
Polygon Names

• 8 sides Octagon
• 9 sides- Nonagon
• 10 sides Decagon
• 12 sides Dodecagon
• 15 sides Pentadecagon
• n sides n-gon

• 3 sides Triangle
• 4 sides Quadrilateral
• 5 sides Pentagon
• 6 sides Hexagon
• 7 sides Heptagon

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Vocabulary
Every segment in the polygon that joins two
non-consecutive vertices is a diagonal.
Interior angles are formed by two consecutive
sides of a polygon. Exterior angles are adjacent
and supplementary to an interior angle of the
polygon.
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Finding Sum of Angles

• To find the number of degrees in a polygon, draw
  all the diagonals possible from one vertex. Then
  count the number of triangles formed and multiply
  that by 180 (the number of degrees in one
  triangle).
• Example When 2 diagonals are drawn in the
  figure below, 3 triangles are formed. In
  conclusion, the sum of the measures of the angles
  in a pentagon is 3(180) or 540.
• But, by using Theorem 55, the sum of the measures
  of the angles could be found in an easier way.

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Theorem 55

• The sum Si of the measures of the angles of a
  polygon with n sides is given by the formula Si
  (n-2)180.
• Example
• What is the sum of the measures of the angles
  in a heptagon?
• Solution Use the formula above and
  substitute
  7 for n
• Si (7-2)180
• (5)180
• 900

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Theorem 56

• If one exterior angel is taken at each vertex,
  the sum Se of the measures of the exterior angles
  of a polygon is given by the formula Se 360.
• Therefore, the sum of the measures of the
  exterior angles in any polygon is 360.

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Theorem 57

• The number of diagonals that can be drawn in a
  polygon of n sides is given by the formula
• d n(n-3)
• 2
• Example
• How many diagonals can be drawn in an 18-gon?
• Solution Use the formula above and
  substitute 18 for n
• d 18(18-3)
• 2
• 270
• 2
• 135 diagonals

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Regular Polygon Formulas

• To find the measure of one angle of a regular
  polygon with n sides, use the following formula
  
• I (n-2)180
• n
• Example
• What is the measure of one angle in a
  regular nonagon?
• Solution Use the formula above and
  substitute 9 for n
• I (9-2)180
• 9
• 1260
• 9
• 140

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Regular Polygon Formulas (Cont.)

• To find the measure of one exterior angle of a
  regular polygon with n sides, use the following
  formula
• E 360
• n
• Example
• What is the measure of one exterior angle of
  a regular octagon?
• Solution Use the formula above and substitute
  8 for n
• E 360
• 8
• 45

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Practice Problems

• How many sides does a polygon have if the sum of
  the measures of its angles is 3240?
• What is the sum of the measures of the angles of
  a 31-gon?
• Given m A 85, m B 115, m C 95,
• m D 100
• Find m E

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Practice Problems (Cont.)

1. What is the sum of the measures of the exterior
   angles, one per vertex, of a decagon?
2. What is the name of a polygon with 65 diagonals?
3. How many diagonals does a 22-gon have?
4. What is the measure of one angle of a regular
   decagon?
5. What regular polygon has an angle measuring 150?
   

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Practice Problems (Cont.)

• 9. What regular polygon has an exterior angle
  measuring 6?
• What is the measure of one exterior angle of a
  regular octagon?
• Answers on next slide

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Answers to Practice Problems

1. 20 sides
2. 5220
3. m E 145
4. 360
5. 13-gon

• 209
• 144
• Dodecagon
• 60-gon
• 45

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Works Cited

• Rhoad, Richard, George Milauskas, and
  Robert Whipple. Geometry for
  Enjoyment and Challenge. Evanston, Illinois
  McDougal, Littell Company, 1991.
• Habeeb, Danielle. Diagonals in a
  Polygon.
• Geometry for Middle School Teachers
  Institute.
• CPTM. 24 May 2008 lthttp//intermath.coe.
• uga.edu/tweb/cptm1/dhabeeb/diagonals/
• diagonalsinapolygon.htmgt.