Thursday February 2
More Puzzles and Polygons
 Tangrams. More
 Tangrams – Use
all seven Chinese puzzle pieces to make shapes
and solve problems.
 Tangram Activity
Use templates to cut out pieces
from larger (blue) sheets. [#Partners=3.]
Geometric Puzzle Foundations
 Measuring angles, lengths and areas.
 Triangles
: add to 180 degrees straight angle [Illustrate physically and with wingeometry]
 Squares, rectangles : 90 degree/ right
angle
 parallelograms: opposite angles are congruent,
sum of consecutive angles =180 degrees
 Dissections, cut and paste methods of measurement.
 Cutting and reassembling polygons.
 The
"Square Me" Puzzle
 The triangle, quadrilateral, pentagon, and hexagon.
More on measurements of angles of polygons with n sides. .
When n = 3 this is a triangle, n=4, a quadrilateral, or
when n= 5, a pentagon.
The sum of the measures of the
interior angles of a triangle is 180 degrees.
What about a quadrilateral? and a pentagon?
or an n sided polygon ( an "n gon")?
 A quadrilateral can be made from two triangles...
so the sum of its interior angles is 2 * 180 = 360.
 A pentagon can be made from 3 triangles...
so the sum of its interior angles is 3* 180 = 540. If the pentagon has
all angles congruent( of equal measurement) then each angle will be 540/5
= 108 degrees!
 A hexagon can be made from 4 triangles...
so the sum of its interior angles is 4* 180 = 720. If the hexagon has all
angles congruent( of equal measurement) then each angle will be 720/6 =
120 degrees!
From the figures we saw that for a quadrilateral (n =4), which can be dissected
into two triangles, the sum is 2*180= 360 degrees.
And for a pentagon (n=5)
which can be dissected into 3 triangles, the sum is 3*180=540 degrees.
# of sides

# of Triangles

Sum of Interior <‘s

If equal, Measure of a single <

3

1

180

60

4

2

360

90

5

3

540

108

6

4

720

120

N

?



In general: the sum
of the interior angles in a n sided polygon
is
(n2) *180 degrees.
A regular polygon
is a polygon where the sides are all of equal length
and the angles are all congruent (or of equal measure).
Question: what is the measure of an individual
angle in a regular polygon with n sides?
For a triangle, the individual angle is 180/3
= 60 degrees.
For a square, the individual angle is 360/4=90
degrees.
For a regular pentagon.... 3*180/5 = 540/5
=108 degrees.
Now for a HEXAGON (6 sides) the sum of the angles
is 720 degrees.
So ... for a REGULAR HEXAGON,
the individual angle is 4*180/6 =720/6 =120 degrees.
In general: The individual
angle for a regular polygon with n sides is (n2)*180/n degrees.
This can be expressed in other ways using algebra:
(n2)*180/n = [180 n  360] / n = 180  360/n.
name of polygon 

degrees of the interior
measure of each angle 
360 degrees divided
by # in Column 2 
equilateral triangle 
3

60 
360 / 3 = 120 
square 
4

90 
360/4= 90

regular pentagon 
5

3*180/5= 108

360/5= 72

regular hexagon 
6

4*180/6=120

360/6= 60

regular heptagon 
7

5*180/7

360/7

regular octagon 
8

6*180/8=135

360/8 = 45

regular dodecagon

12

10*180/12=1800/12=150

360/12=30

Cutting and reassembling polygons.
Convex: Any two points in the figure
have a line segment connecting them. If that line segment is always inside the figure, then the figure
is called "convex".
Making Dissection Puzzles [more links] :
Dissection of the plane Tilings
of the plane:
Tessellations
 Activity: Tile plane with playing cards.
 FAPP watch Video: Possible tiles and tilings.
 One Regular Convex Polygon.
 Two or more Regular Convex Polygons.
 Other Polygon Tilings.
 Non polygonal tilings.
 One polygonal Tile: Quadrilateral Activity.

An online tool for making tilings of the plane
 Regular and semiregular Tilings of the plane. Some explorations  classification.
 A tiling is a regular tiling if (i)it has a single tile shape that
is a regular polygon and (ii) the vertices and edges of the tiles coincide
(no overlapping edges)
 A tiling is a semiregular tiling if (ii) each tile shape is a regular
polygon, (ii) the vertices and edges of the tiles coincide (no overlapping
edges) and (iii) every vertex has the same polygon types arranged around
it.
 Wingeometry download!
and demonstrate tesselations.
 Naming tilings
(Math Forum)
 The numbers represent the number of sides in the poygons.
 The order indcates the order in which the poygons are arranged about
a vertex.
 Semiregular Tilings: global results!
Look at the results using wingeometry.
 Student lesson
(Math Forum) a place for further explorations online.