11.1 Angle Measures in Polygons

Sum of measures of interior angles # of triangles Sum of measures of interior angles # of sides 1(180)=180 3 1 2(180)=360 4 2 3 3(180)=540 5 6 4 4(180)=720 n-2 (n-2) • 180 n

If a convex polygon has n sides, then the sum of the measure of the interior angles is (n – 2)(180°)

If a regular convex polygon has n sides, then the measure of one of the interior angles is

Ex. 1 Use a regular 15-gon to answer the questions. Find the sum of the measures of the interior angles. Find the measure of ONE interior angle 2340° 156°

x = 104 Ex: 2 Find the value of x in the polygon x 126 100 143 130 117

Ex: 3 The measure of each interior angle is 150°, how many sides does the regular polygon have? One interior angle A regular dodecagon

Two more important terms Interior Angles Exterior Angles

1 2 3 4 5 The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360°.

The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360°. 1 3 2

The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360°. 1 2 4 3

The measure of each exterior angle of a regular polygon is

Ex. 4 Find the measure of ONE exterior angle of a regular 20-gon. 18°

Ex. 5 Find the measure of ONE exterior angle of a regular heptagon. 51.4°

Ex. 6 The sum of the measures of five interior angles of a hexagon is 625. What is the measure of the sixth angle? 95°

Let’s practice! 11.1 Worksheet

11.2 Area of Regular Polygons

Area of an Equilateral Triangle 30 30 s s 60 60 s

Ex: 1 Find the area of an equilateral triangle with 4 ft sides.

A Circle can be circumscribed around any regular polygon

VERTICES

A Central Angle is an angle whose vertex is the center and whose sides are two consecutive radii A RADIUS joins the center of the regular polygon with any of the vertices

How many equilateral triangles make up a regular Hexagon? Equal Sides s Equal Angles How many equilateral triangles make up a regular Hexagon? What is the area of each triangle? What is the area of the hexagon? 6 • (the area of the triangle)

41.569 units2 What is the area of this regular hexagon? 4 The area of an equilateral triangle A = 6.9282 The area of our equilateral triangle in this example How many identical equilateral triangles do we have? 6 A = 6 * (6.9282) The area of our hexagon in this example

An APOTHEM is the distance between the center and a side An APOTHEM is the distance between the center and a side. (It MUST be perpendicular to the side.)

You need to know the apothem and perimeter How to find the Area of ANY REGULAR POLYGON You need to know the apothem and perimeter Area = (1/2)•a•P or A = .5•a•P

A = ½ aP a Area of a Regular Polygon: A = .5 (apothem) (# of sides)(length of each side) a

A Regular Octagon 7 ft

360/8=45 22.5° 45 x 7 3.5 ft

Area = 236.6 ft2 Area = .5 • 8.45 • 56 Perimeter is 56 feet Apothem is 8.45 feet 7 ft What is the area? Area = .5 • 8.45 • 56 Area = 236.6 ft2

11.2 Worksheet Practice B ODDS Let's Practice 11.2 Worksheet Practice B ODDS

Homework Worksheets’ EVENS