Download Polygons_worksheet3 - Penns Valley Math Resources

Student Material
Geometry/angles in polygons
Christopher Yakes, GK-12 Fellow, 2003-2004
UCLA Science and Mathematics Inquiry
Polygons
Polygons can be found almost
everywhere in the world, from natural rock
formations to 3D animation using
computers to crystal formations.
Mathematicians are naturally curious people. Since
polygons are all over the place, it is not a surprise that
mathematicians have been studying polygons for
centuries.
One thing we have to be able do in order to work with polygons is to name them.
Here is a table showing the names of various polygons:
Number of sides
Name
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
n
n-gon
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Student Material
Geometry/angles in polygons
Christopher Yakes, GK-12 Fellow, 2003-2004
UCLA Science and Mathematics Inquiry
EXERCISE 1: Can you find a triangle in the picture below? A quadrilateral? A
hexagon? Can you find any other polygons in the picture?
Pompeii, Italy
The straight lines that make up the sides of a polygon are called EDGES. The
corner that two edges of a polygon make is called a VERTEX (plural vertices).
The angle made at the vertex of a polygon is called an INTERIOR ANGLE.
VERTEX
EDGE
INTERIOR ANGLES
The polygons we work with are usually CONVEX polygons. That means that if
you draw a line from one vertex to another, it doesn’t cross any edges, or lie on
the exterior of the polygon.
NOT CONVEX
CONVEX
When a polygon is not convex we say it is CONCAVE.
EXERCISE 2: Draw a convex nonagon. Draw a concave heptagon.
Page 2 of 5
Student Material
Geometry/angles in polygons
Christopher Yakes, GK-12 Fellow, 2003-2004
UCLA Science and Mathematics Inquiry
If we want to show that two edges of a polygon have the same length then we
put small dashes in the two edges. We say the edges are congruent.
When we want to show that two interior angels have the same
measure then we put small curves on them as shown.
If a polygon has edges all congruent, we say it is
equilateral.
If a polygon has all interior angles congruent, we say it is
equiangular.
If a polygon has all edges and all vertices congruent, we
say it is regular.
Drawings of some regular
polygons are shown on the
right.
Notice that the measure of
one vertex is given in each
of the pictures.
Since the polygons are
regular, all the vertices
have the same measure.
Take for example the
hexagon. It has six interior
angels, and the measure of
each one is 120º.
That means that the sum of
the interior angles of a
regular hexagon is 6120º
= 720º.
In fact, this is true of any
hexagon!
In general, remember that the sum of the interior angles of a polygon is
(n 2)180º
where n is the number of sides.
Page 3 of 5
Student Material
Geometry/angles in polygons
Christopher Yakes, GK-12 Fellow, 2003-2004
UCLA Science and Mathematics Inquiry
EXERCISE 3: What is the sum of the interior angles of a 13-gon?
EXERCISE 4: Can the sum of the interior angles of a polygon be 450º?
EXERCISE 5: What is the measure of the interior angle of a regular
dodecahedron (12 sided polygon)?
Many interesting
shapes are formed
using regular
polygons.
Don’t forget that we often label
the vertices of a polygon
using letters of the alphabet:
The picture shows a hexagon
with vertices labeled A through F.
We would call it
B
A
C
F
E
D
Page 4 of 5
hexagon ABCDEF
Student Material
Geometry/angles in polygons
Christopher Yakes, GK-12 Fellow, 2003-2004
UCLA Science and Mathematics Inquiry
Teacher Tips:
1. Have students read the previous worksheets aloud one paragraph at a
time, and work the examples out in class together.
2. Great lesson to use as a review for a midterm on polygons.
3. More facts about polygons could also be added.
Page 5 of 5