2. Essential Questions
✤ How are polygons classified according to their sides?
✤ How do you find the sum of the angle measures of polygons?
✤ Where you’ll see this:
✤ Safety, hobbies, nature
4. Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
their endpoints
2. Side:
3. Vertex:
4. Convex:
5. Concave:
6. Regular Polygon:
7. Diagonal:
5. Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
their endpoints
2. Side: One of the segments that makes up the polygon
3. Vertex:
4. Convex:
5. Concave:
6. Regular Polygon:
7. Diagonal:
6. Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
their endpoints
2. Side: One of the segments that makes up the polygon
3. Vertex: The point where segments meet
4. Convex:
5. Concave:
6. Regular Polygon:
7. Diagonal:
7. Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
their endpoints
2. Side: One of the segments that makes up the polygon
3. Vertex: The point where segments meet
4. Convex: When there are no indentations in a polygon
5. Concave:
6. Regular Polygon:
7. Diagonal:
8. Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
their endpoints
2. Side: One of the segments that makes up the polygon
3. Vertex: The point where segments meet
4. Convex: When there are no indentations in a polygon
5. Concave: When there is an indentation into a polygon
6. Regular Polygon:
7. Diagonal:
9. Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
their endpoints
2. Side: One of the segments that makes up the polygon
3. Vertex: The point where segments meet
4. Convex: When there are no indentations in a polygon
5. Concave: When there is an indentation into a polygon
6. Regular Polygon: A polygon where all the sides and angles are
congruent
7. Diagonal:
10. Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
their endpoints
2. Side: One of the segments that makes up the polygon
3. Vertex: The point where segments meet
4. Convex: When there are no indentations in a polygon
5. Concave: When there is an indentation into a polygon
6. Regular Polygon: A polygon where all the sides and angles are
congruent
7. Diagonal: A segment that joins two vertices but is not a side
11. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
8 sides: 9 sides: 10 sides:
12. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
Pentagon
8 sides: 9 sides: 10 sides:
13. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
Pentagon
8 sides: 9 sides: 10 sides:
14. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
Pentagon Hexagon
8 sides: 9 sides: 10 sides:
15. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
Pentagon Hexagon
8 sides: 9 sides: 10 sides:
16. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
Pentagon Hexagon Heptagon
8 sides: 9 sides: 10 sides:
17. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
Pentagon Hexagon Heptagon
8 sides: 9 sides: 10 sides:
25. Polygons and Their Sides
5 sides: 6 sides: 7 sides:
Pentagon Hexagon Heptagon
8 sides: 9 sides: 10 sides:
Octagon Nonagon Decagon
Anything larger: n-gon, where n is the number of sides
26. Example 1
Name each polygon by its number of sides and label as concave or
convex.
27. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave
28. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave
Pentagon
29. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave Convex
Pentagon
30. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave Convex
Pentagon Octagon
31. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave Convex
Pentagon Octagon
Convex
32. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave Convex
Pentagon Octagon
Convex
Quadrilateral
33. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave Convex
Pentagon Octagon
Convex Concave
Quadrilateral
34. Example 1
Name each polygon by its number of sides and label as concave or
convex.
Concave Convex
Pentagon Octagon
Convex Concave
Quadrilateral Nonagon
35. # of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
36. # of sides: 3 # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
37. # of sides: 3 # of sides:
# of triangles: 1 # of triangles:
Degrees: Degrees:
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
38. # of sides: 3 # of sides:
# of triangles: 1 # of triangles:
Degrees: 180° Degrees:
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
39. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles:
Degrees: 180° Degrees:
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
40. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles:
Degrees: 180° Degrees:
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
41. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees:
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
42. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
43. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
44. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
45. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides:
# of triangles: # of triangles:
Degrees: Degrees:
46. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides:
# of triangles: 3 # of triangles:
Degrees: Degrees:
47. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides:
# of triangles: 3 # of triangles:
Degrees: 540° Degrees:
48. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides: 6
# of triangles: 3 # of triangles:
Degrees: 540° Degrees:
49. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides: 6
# of triangles: 3 # of triangles:
Degrees: 540° Degrees:
50. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides: 6
# of triangles: 3 # of triangles:
Degrees: 540° Degrees:
51. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides: 6
# of triangles: 3 # of triangles:
Degrees: 540° Degrees:
52. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides: 6
# of triangles: 3 # of triangles: 4
Degrees: 540° Degrees:
53. # of sides: 3 # of sides: 4
# of triangles: 1 # of triangles: 2
Degrees: 180° Degrees: 360°
# of sides: 5 # of sides: 6
# of triangles: 3 # of triangles: 4
Degrees: 540° Degrees: 720°
54. Angle Sum of a Polygon:
Angle Measure of a Regular Polygon:
55. Angle Sum of a Polygon: The sum of the interior angles of a polygon
with n sides is given by the formula
Angle Measure of a Regular Polygon:
56. Angle Sum of a Polygon: The sum of the interior angles of a polygon
with n sides is given by the formula
Angle Measure of a Regular Polygon: The measure of each interior
angle of a regular polygon with n sides is given by the formula
(n − 2)180°
S=
n
57. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
58. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
F B
E C
D
59. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
F B
10x
E C
D
60. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
F B
10x
3x + 8
E C
3x + 8
D
61. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
F B
10x
7x - 22 3x + 8
E C
3x + 8
D
62. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
F B
8x - 12 10x
7x - 22 3x + 8
E C
3x + 8
D
63. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
S = (n − 2)180°
F B
8x - 12 10x
7x - 22 3x + 8
E C
3x + 8
D
64. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
S = (n − 2)180°
F B
8x - 12 10x S = (6 − 2)180°
7x - 22 3x + 8
E C
3x + 8
D
65. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
S = (n − 2)180°
F B
8x - 12 10x S = (6 − 2)180°
S = (4)180°
7x - 22 3x + 8
E C
3x + 8
D
66. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
S = (n − 2)180°
F B
8x - 12 10x S = (6 − 2)180°
S = (4)180°
7x - 22 3x + 8 S = 720°
E C
3x + 8
D
67. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
S = (n − 2)180°
F B
8x - 12 10x S = (6 − 2)180°
S = (4)180°
7x - 22 3x + 8 S = 720°
E C
3x + 8 The sum of all of the angles is 720°
D
68. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
F B
8x - 12 10x
7x - 22 3x + 8
E C
3x + 8
D
69. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
F B
8x - 12 10x
7x - 22 3x + 8
E C
3x + 8
D
70. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
F B
8x - 12 10x
7x - 22 3x + 8
E C
3x + 8
D
71. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
F B
8x - 12 10x
41x − 18 = 720°
7x - 22 3x + 8
E C
3x + 8
D
72. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
F B
8x - 12 10x
41x − 18 = 720°
+18 +18
7x - 22 3x + 8
E C
3x + 8
D
73. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
F B
8x - 12 10x
41x − 18 = 720°
+18 +18
41x = 738
7x - 22 3x + 8
E C
3x + 8
D
74. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
F B
8x - 12 10x
41x − 18 = 720°
+18 +18
41x = 738
41 41
7x - 22 3x + 8
E C
3x + 8
D
75. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
F B
8x - 12 10x
41x − 18 = 720°
+18 +18
41x = 738
41 41
7x - 22 3x + 8
E C x = 18
3x + 8
D
76. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x
7x - 22 3x + 8
E C
3x + 8
D
77. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) =
7x - 22 3x + 8
E C
3x + 8
D
78. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180°
7x - 22 3x + 8
E C
3x + 8
D
79. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
7x - 22 3x + 8
E C
3x + 8
D
80. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 =
7x - 22 3x + 8
E C
3x + 8
D
81. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62°
7x - 22 3x + 8
E C
3x + 8
D
82. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62° = m∠C = m∠D
7x - 22 3x + 8
E C
3x + 8
D
83. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62° = m∠C = m∠D
7(18) - 22 =
7x - 22 3x + 8
E C
3x + 8
D
84. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62° = m∠C = m∠D
7(18) - 22 = 104°
7x - 22 3x + 8
E C
3x + 8
D
85. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62° = m∠C = m∠D
7(18) - 22 = 104° = m∠E
7x - 22 3x + 8
E C
3x + 8
D
86. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62° = m∠C = m∠D
7(18) - 22 = 104° = m∠E
7x - 22 3x + 8
E C
8(18) - 12 =
3x + 8
D
87. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62° = m∠C = m∠D
7(18) - 22 = 104° = m∠E
7x - 22 3x + 8
E C
8(18) - 12 = 132°
3x + 8
D
88. Example 2
In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
the angles of the hexagon, then find the measure of each angle.
A
10x
x = 18
F B
8x - 12 10x 10(18) = 180° = m∠A = m∠B
3(18) + 8 = 62° = m∠C = m∠D
7(18) - 22 = 104° = m∠E
7x - 22 3x + 8
E C
8(18) - 12 = 132° = m∠F
3x + 8
D
89. Example 3
Find the measure of each angle of a regular 14-gon.
90. Example 3
Find the measure of each angle of a regular 14-gon.
(n − 2)180°
S=
n
91. Example 3
Find the measure of each angle of a regular 14-gon.
(n − 2)180°
S=
n
(14 − 2)180°
S=
14
92. Example 3
Find the measure of each angle of a regular 14-gon.
(n − 2)180°
S=
n
(14 − 2)180°
S=
14
(12)180°
S=
14
93. Example 3
Find the measure of each angle of a regular 14-gon.
(n − 2)180°
S=
n
(14 − 2)180°
S=
14
(12)180°
S=
14
2160°
S=
14
94. Example 3
Find the measure of each angle of a regular 14-gon.
(n − 2)180°
S=
n
(14 − 2)180°
S=
14
S = 154 2 7 °
(12)180°
S=
14
2160°
S=
14