Name polygons based on their number of sides Classify polygons Calculate and use the measures of interior and exterior angles

1. Properties of Polygons
The student is able to (I can):
• Name polygons based on their number of sides
• Classify polygons based on
– concave or convex
– equilateral, equiangular, regular
• Calculate and use the measures of interior and exterior
angles of polygons

2. polygonpolygonpolygonpolygon – a closed plane figure formed by three or more
noncollinear straight lines that intersect only at their
endpoints.
polygons
not
polygons

3. vertexvertexvertexvertex – the common endpoint of two sides. Plural:
verticesverticesverticesvertices.
diagonaldiagonaldiagonaldiagonal – a segment that connects any two nonconsecutive
vertices.
regularregularregularregular – a polygon that is both equilateral and equiangular.
vertex
diagonal

4. Polygons are named by the number of their sides:
SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.
Each Int.Each Int.Each Int.Each Int.
(Regular)(Regular)(Regular)(Regular)
3333 Triangle
4444 Quadrilateral
5555 Pentagon
6666 Hexagon
7777 Heptagon
8888 Octagon
9999 Nonagon
10101010 Decagon
12121212 Dodecagon
nnnn n-gon

5. Examples: Identify the general name of each polygon:
1.
2.
3.

6. Examples: Identify the general name of each polygon:
1.
2.
3.
pentagon
dodecagon
quadrilateral

7. concaveconcaveconcaveconcave – a diagonal of the polygon contains points outside
the polygon. (“caved in”)
convexconvexconvexconvex – not concave.
concave
pentagon
convex
quadrilateral
outside the
polygon

8. We know that the angles of a triangle add up to 180°, but
what about other polygons?
Draw a convex polygon of at least 4 sides:

9. We know that the angles of a triangle add up to 180°, but
what about other polygons?
Draw a convex polygon of at least 4 sides:
Now, draw all possible diagonals from one vertex. How many
triangles are there?
What is the sum of their angles?
180°
180°
180°

10. We know that the angles of a triangle add up to 180°, but
what about other polygons?
Draw a convex polygon of at least 4 sides:
Now, draw all possible diagonals from one vertex. How many
triangles are there? 3333
What is the sum of their angles? 3(180) = 5403(180) = 5403(180) = 5403(180) = 540°°°°
180°
180°
180°

11. Polygon Angle Sum Theorem
The sum of the interior angles of a convex polygon with n
sides is
(n – 2)180°
If the polygon is equiangular, then the measure of one
angle is
− °( 2)180n
n

12. Let’s fill out the rest of the table:
SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.
Each Int.Each Int.Each Int.Each Int.
(Regular)(Regular)(Regular)(Regular)
3333 Triangle 1 (1)180°=180° 60°
4444 Quadrilateral 2 (2)180°=360° 90°
5555 Pentagon 3 (3)180°=540° 108°
6666 Hexagon
7777 Heptagon
8888 Octagon
9999 Nonagon
10101010 Decagon
12121212 Dodecagon
nnnn n-gon

13. Let’s fill out the rest of the table:
SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.
Each Int.Each Int.Each Int.Each Int.
(Regular)(Regular)(Regular)(Regular)
3333 Triangle 1 (1)180°=180° 60°
4444 Quadrilateral 2 (2)180°=360° 90°
5555 Pentagon 3 (3)180°=540° 108°
6666 Hexagon 4 (4)180°=720° 120°
7777 Heptagon 5 (5)180°=900° ≈128.6°
8888 Octagon 6 (6)180°=1080° 135°
9999 Nonagon 7 (7)180°=1260° 140°
10101010 Decagon 8 (8)180°=1440° 144°
12121212 Dodecagon 10 (10)180°=1800° 150°
nnnn n-gon n – 2 (n – 2)180°
( 2)180n
n
− °

14. Recall that an exterior angle is an angle created by extending
the side of a polygon:
Now, consider the exterior angles of a regular pentagon:
Exterior
angle

15. From our table, we know that each interior angles is 108°.
This means that each exterior angle is 180 – 108 = 72°.
The sum of the exterior angles is therefore 5(72) = 360°. It
turns out this is true for any convex polygon, regular or not.
108°
72°
72°
72°
72°
72°

16. Polygon Exterior Angle Sum Theorem
The sum of the exterior angles of a convex polygon is
360°.
For any equiangular convex polygon with n sides, each
exterior angle is
360
n
°
SidesSidesSidesSides NameNameNameName Sum Ext.Sum Ext.Sum Ext.Sum Ext. Each Ext.Each Ext.Each Ext.Each Ext.
3333 Triangle 360° 120°
4444 Quadrilateral 360° 90°
5555 Pentagon 360° 72°
6666 Hexagon 360° 60°
8888 Octagon 360° 45°
nnnn n-gon 360° 360°/n