Tuesday, February 1

Discuss Project proposal... Guidelines are on web! Form partnerships?

Dissection of the plane--- Tilings of the plane:
Tessellations


  • More on measurements of angles and areas of polygons.
  •    
    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.