Download Polygons

Polygons
Polygons and Classifying Polygons
Polygons: Any shape with straight edges that is closed and has
more than three SIDES. Each endpoint is a VERTEX of the
B
polygon.
A
C
D
E
Polygon
NOT a Polygon
Each Polygon must be CLOSED
(no gaps between the sides)
To name a polygon, start at any vertex and list the vertices consecutively in a
clockwise or counter clockwise direction.
The name of the polygon above is ABCDE, BCDEA, DCBAE, etc.
Ex. 1 Name the polygon. Then identify its vertices, sides, and
angles.
X
Y
Quadrilateral:
any polygon that has 4 sides
Z
U
Polygon Name: XYUZ
Vertices: X, Y, Z, U
Sides: XY, YU, UZ, ZX
Angles:  X,  Y,  Z,  U
Ex. 2
Is the figure a polygon? Explain.
a)
b)
Yes, polygon is formed by 5
straight line segments
c)
No, it is not “closed”
No, all sides are not line
segments
d)
Yes, polygon is formed by 7
straight line segments
Classifying Polygons
You can classify a polygon by the number of sides it has.
The table below shows the names of some common polygons.
Sides
Name
3
Triangle
4
Quadrilateral
5
Pentagon
6
7
Hexagon
Heptagon
8
Octagon
9
Nonagon
10
11
Decagon
12
N
Undecagon
Dodecagon
N - gon
Convex or Concave You can classify polygons as Convex or
Concave:
• Convex Polygon: No line that contains a side of the polygon passes
through the interior of the polygon.
• Concave Polygon: At least one line extended side passes through
the interior.
Convex
Concave
Ex. 3 Classify the polygon by its sides.
Identify each as convex or concave.
Pentagon
Convex
Decagon
Concave
• A polygon is equilateral, if all the sides are congruent.
• A polygon is equiangular, if all its interior angles are congruent.
• A polygon is regular if it is both equilateral and equiangular
Equilateral Polygon
Equiangular Polygon
Regular Polygon
Ex. 4 The polygon is equiangular.
Find the value of x.
6xo
240o
6x = 240
x = 240
6
x = 40o
Equiangular polygon
Ex. 5 Polygon is regular. Find the value of x.
7x - 1
20
7x - 1 = 20
7x = 21
x = 21
7
x = 3 units
Regular polygon
Class Work
• Pg 306 #3– 5, #8 – 10
• Pg 413 – 415 #2 – 21, 27, 36 – 38
Interior Angles and Exterior Angles
Polygon Interior Angles Theorem:
The sum of the measure of the interior angles of
a convex polygon with N sides is
o
(n
–
2)
∙
180
_____________
Ex. 1 Determine the sum of the interior angles of a
a.
16 – sided polygon
(16 – 2) ∙ 180o
= (14) ∙ 180o
= 2520o
b.
Poly. int. angles thm
25 – sided polygon
(25 – 2) ∙ 180o
= (23) ∙ 180o
= 4140o
Poly. int. angles thm
Ex. 2 How many sides does a polygon if the sum of
its interior angles is 3420o?
(n – 2) ∙ 180 = 3420
(n – 2) = 3420
180
(n – 2) = 19
n = 19 + 2
n = 21 sides
Polygon interior angles
theorem
Ex. 3 Write an equation for the sum of the interior angles.
Then solve for x.
Determine the sum of the interior
angles.
Polygon interior angles theorem
(4 – 2) ∙ 180o
= (2) ∙ 180o
= 360o
Solve for x.
(2x – 15) + x + (2x – 15) + x = 360
6x – 30 = 360
6x = 390
x = 390__
6
x = 65
Polygon Exterior Angles Theorem:
The sum of the measures of the
exterior angles of a convex polygon is
always __________
360o
A
B
C
D
E
Ex. 4
Find the measure of an exterior angle for each regular convex
polygon.
a) 20 sides
let x = measure of the exterior angle
20x = 360 Poly ext angles thm
x = __360___ x = 18o
20
b) 15 sides
let x = measure of the exterior angle
15x = 360 Poly ext angles thm
x = __360___
15
x = 24o
Ex. 5 Determine the value of x.
105 + 8x + 9x = 360
105 + 17x = 360
17x = 360 - 105
17x = 255
x = __255___
17
x = 15
Poly ext angles thm
Class Work
• Page 418 #2,4
• Page 420 #4-7
• Page 421 #8–24 (even)
• Page 423 #1-8