Polygons
Polygons are old business. We also took a look at tessellations last
year. So, complete these definitions. Use your textbook as your
resource.
Polygon
Regular Polygon
Tesselation
Concave Polygon
Convex Polygon
Given a shape you should be able to tell if it is, indeed, a polygon.
If the shape is a polygon you should be able to classify it by the
number of sides it has. In addition, you should be able to describe
it as regular or not regular. And, last of all, you should be able to
decide if it, on its own, could be used to create a tessellation.
Here are some shapes. Talk about them and record what you
observe.
Next, we will focus on the angles inside the polygon. Though we will use regular
shapes, keep in mind that the total of the degrees of all the interior angles is the
same for that group of polygons. When the shape is a regular polygon, we can
determine the measure of a single angle by dividing by the number of angles in the
shape.
Polygon
Sides
Sum of Inside Angles One angle
Tessellates?
n-gon
So, if we are working with a regular 16-gon, what is the measure
of each of its interior angles? Answer to the nearest tenth of a
degree if rounding is needed.
Now, let’s take a peek at the exterior angles on a polygon.
Something almost magical happens. What is the sum of the
measures of the exterior angles of any polygon?
We’ll use this shape to demonstrate.
It is a heptagon that is convex and not
regular.
Now, let’s think about tessellations and complete the chart with a
YES for “the polygon tessellates” and a NO for “the polygon does
not tessellate.” How do we make that decision? What must be the
sum of the measures of the angles at any corner of a
tessellation?
Here is a part of a tessellation
Made using different polygons.
Look closely at each intersection
Of sides of the polygons. What
Is the sum of the measures of
The angles at ecah intersection?
Now you can see why we often mix polygons when creating a
tessellation. Some polygons need help from others to tessellate!
NAME ______________________________________________ DATE ______________ PERIOD _____
10-6 Practice
Polygons
Find the sum of the measures of the interior angles of each polygon.
1. quadrilateral 360°
2. decagon 1440°
3. 12-gon 1800°
4. heptagon 900°
5. pentagon 540°
6. hexagon 720°
7. 25-gon 4140°
8. 100-gon 17,640°
Find the measure of an interior angle of each polygon.
9. regular nonagon 140°
12. regular 12–gon 150°
10. regular octagon 135°
11. regular hexagon 120°
13. regular quadrilateral 90° 14. regular decagon 144°
TESSELLATIONS For Exercises 15 and 16, identify the polygons used to create
each tessellation.
15.
16.
triangles, quadrilaterals,
hexagons, octagons
triangles, squares,
hexagons, octagons
17. Which figure best represents a regular polygon? D
A
©
Glencoe/McGraw-Hill
B
C
576
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Glencoe Pre-Algebra
NAME ______________________________________________ DATE ______________ PERIOD _____
10-6 Enrichment
Polygons and Diagonals
A diagonal of a polygon is any segment that connects two nonconsecutive vertices of
the polygon. In each of the following polygons, all possible diagonals are drawn.
Examples
I
D
A
B
C
In ABC, no diagonals
can be drawn. Why?
E
G
F
In quadrilateral DEFG,
2 diagonals can be drawn.
H
J
L
K
In pentagon HIJKL,
5 diagonals can be
drawn.
Complete the chart below and try to find a pattern that will help you answer the
questions that follow.
Number of Sides
Number of Diagonals
From One Vertex
Total Number
of Diagonals
triangle
3
0
0
quadrilateral
4
1
2
pentagon
5
2
5
1.
hexagon
6
2.
heptagon
7
3.
octagon
8
4.
nonagon
9
5.
decagon
10
3
4
5
6
7
9
14
20
27
35
Polygons
Find the total number of diagonals that can be drawn in a polygon with the given
number of sides.
6. 6 9
7. 7 14
8. 8 20
9. 9 27
10. 10 35
11. 11 44
12. 12 54
13. 15 90
14. 20 170
15. 50 1175
16. 75 2700
17. n ©
Glencoe/McGraw-Hill
578
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2
Glencoe Pre-Algebra