11 Area of Regular Polygons and Circles

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Presentation transcript:

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 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.

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°.

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

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 °

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°

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.

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.

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°

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

Your turn again. How many sides does a polygon have if each interior angle has… 165.6° 25 sides 162° 20 sides 120° 6 sides 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

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