Thursday  September 15





REVIEW: Recall our previous discussions for a polygon 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.
Review Question:What about a quadrilateral? and a pentagon?  or an n sided polygon  ( an "n -gon")?

 

From the figure 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.



In general: the sum of the interior angles in a n sided polygon is
                                      (n-2) *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).



Review 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 (n-2)*180/n degrees.
This can be expressed in other ways using algebra:
(n-2)*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



SO....
180/360 = 1/n + 1/k + 1/p  or
1/n + 1/k + 1/p=1/2
1/3+1/4+1/5 >1/2.