11 Area of Regular Polygons and Circles

11.1 Angles in Polygons

Quick ReviewPolygon—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 PolygonsSum 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: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.

Find the measure of xHow 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?

5Sum of Angles:?(5-2)180 = 540

25x + 40 = 540 25x = 500 x = 20102° , 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-gon15 sides(15-2)180° = 2340°

2340°/15 = 156°

There are 156° in each interior angle of a 15-gon.

Finding the Number of sidesIf 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— ≈ 128.6Dodecagon— 150°

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°

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

HomeworkVocab: 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