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11 Area of Regular Polygons and Circles

11 Area of Regular Polygons and Circles. 11.1 Angles in Polygons. Quick Review. Polygon—any closed figure with straight sides that do not cross one another Regular Polygon—a polygon with all congruent sides and all congruent angles Vertex—Where two sides of a polygon meet

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11 Area of Regular Polygons and Circles

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  1. 11 Area of Regular Polygons and Circles 11.1 Angles in Polygons

  2. Quick Review • Polygon—any closed figure with straight sides that do not cross one another • Regular Polygon—a polygon with all congruent sides and all congruent angles • Vertex—Where two sides of a polygon meet • Interior angle—angle formed on the inside of a polygon by two adjacent sides • Exterior angle—angle formed on the exterior of a polygon by a vertex and a line drawn from the vertex.

  3. Angles in Polygons • Sum of Interior Angles: • (n-2)180° = Sum of Int. < • Where “n” is the number of sides in the polygon. • (the number of sides will be the same as the number of angles) • Sum of Exterior Angles • The sum of the Ext. <‘s in any polygon is 360°.

  4. Example: • Find the sum of the interior angles in a convex Hexagon. • Hexagon is 6-sided. • (6-2)180 ° = 720° • How many sides does a polygon have if each exterior angle measures 45 °? • 360/45 = 8 • This is an 8-sided polygon, aka Octagon.

  5. Find the measure of x • How many sides? • 6 • Sum of Interior Angles? • (6-2)180 ° = 720 ° • The sum of the given angles: • 120 +90+110+130+160=610 • 720 ° -610 ° = 110 ° • x = 110 °

  6. Find the value of x and the measure of each angle. • How many sides? • 5 • Sum of Angles:? • (5-2)180 = 540 • 25x + 40 = 540 • 25x = 500 • x = 20 • 102° , 65°, 168°, 95°, 110°

  7. Find the value of each interior angle in each Regular Polygon… • A Nonagon. • 9-sides • (9-2)180° = 1260° • 1260°/9 = 140° • There are 140° in each interior angle of a Nonagon. • A 15-gon • 15 sides • (15-2)180° = 2340° • 2340°/15 = 156° • There are 156° in each interior angle of a 15-gon.

  8. Finding the Number of sides • If you are given one interior angle of a regular polygon, you can use that info to find the number of sides. • Int. Angle = 160° • Ext. Angle = 180 ° - 160 ° = 20° • 360 ° /20 ° = 18 • There are 18 sides in this polygon.

  9. Your Turn: • Find the sum of the measures of all interior angles of the following: • Decagon— • 1440° • Heptagon— • 900° • Dodecagon— • 1800° • Find the measure of each interior angle in a Regular – • Decagon— • 144° • Heptagon— • ≈ 128.6 • Dodecagon— • 150°

  10. Your Turn, Exterior Angles: • Find the Sum of the Exterior Angles in a: • Decagon • 360° • Heptagon • 360° • Dodecagon • 360° • Find the measure of each Exterior Angle of a Regular … • Decagon • 36° • Heptagon • ≈ 51.4° • Dodecagon • 30°

  11. Your turn again. • How many sides does a polygon have if each interior angle has… • 165.6° • 25 sides • 162° • 20 sides • 120° • 6sides • How many sides does a regular polygon have if each exterior angle has a measure of… • 20° • 18 sides • 40° • 9 sides • 15° • 24 sides

  12. Homework • Vocab: Know the names and number of sides of all the polygons from triangle to dodecagon (except the 11-gon) • Pg. 665-668 #6-25 All; 49-54 All; 58-61 All

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