{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 
128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 
0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
264 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 
0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 128 1 0 0 0 
0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 
}{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 
1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 
8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time
s" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 
2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 50 "High School Modul
es > Geometry by Gregory A. Moore" }{TEXT 267 1 "\n" }}{PARA 3 "" 0 "
" {TEXT -1 4 "    " }{TEXT 256 24 "Regular Polygons & Stars" }}{PARA 
0 "" 0 "" {TEXT -1 161 "\nAn exploration of regular polygons and their
 features - including graphs of the different types of axes of symmetr
y, and a peek at stars of the geometry world.\n" }}{PARA 0 "" 0 "" 
{TEXT 258 153 "[Directions : Execute the Code Resource section first. \+
Although there will be no output immediately, these definitions are us
ed later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 7 "0. Code" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart; with(plots): with(geometry
): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "sbl := 'axes = none
,   scaling = constrained, \n       thickness = 3, color = COLOR(RGB, \+
.5,.4,.8)':\nsgy := 'axes = none,   scaling = constrained, \n       th
ickness = 3, color = COLOR(RGB, .5,.5,.6)':\nsbe := 'axes = none,   sc
aling = constrained, \n       thickness = 3, color = COLOR(RGB, .8,.7,
.5)':\nsor := 'axes = none,   scaling = constrained, \n       thicknes
s = 3, color = COLOR(RGB, .9,.7,.1)':\nsgn := 'axes = none,   scaling \+
= constrained, \n       thickness = 3, color = COLOR(RGB, .4,.7,.4)':
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "cbl := 'color = COLOR(
RGB, .5,.4,.8)':\ncgy := 'color = COLOR(RGB, .5,.5,.6)':\ncbe := 'colo
r = COLOR(RGB, .8,.7,.5)':\ncor := 'color = COLOR(RGB, .9,.7,.1)':\ncy
l := 'color = COLOR(RGB, .9,.8,.1)':\ncgn := 'color = COLOR(RGB, .5,.8
,.5)':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "scanft := 'scali
ng = constrained, axes = none':\nscanft := 'filled = true, axes = none
, scaling = constrained':\nscant2 := 'thickness = 2, axes = none, scal
ing = constrained':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "point(o,0,0);\ncircle( UC ,[
o,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "RegPolyPlot  := \+
 proc( N )\n  local k, n, ngon, c;\n  ngon := n -> [seq([ cos(2*Pi*i/n
), sin(2*Pi*i/n) ], i = 1..n)]:  \n  c := n -> COLOR(RGB, .2+n/(2*N), \+
.2+n/(2*N), .5);\n  display([ seq( polygonplot(ngon(n), color=c(n)), n
 = 3..N)],\n           scaling = constrained, axes = none);\nend proc:
\n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "1. Polyg
ons" }}{PARA 0 "" 0 "" {TEXT -1 67 "\nA polygon is a any simple closed
 shape formed with line segments.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "plot( [ [0,0],[2,0],[3,5],[1,4],[-2,1],[0,0]], sgy );
" }}}{PARA 0 "" 0 "" {TEXT -1 46 "\n\nSome familiar polygons include t
riangles ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "display( pl
ot( [ [3,0],[7,0],[5,6],[3,0]], sgy ),\n         plot( [ [-7,5],[-5,7]
,[-2,5],[-7,5]], sbe ), \n         plot( [ [-5,3],[3,3],[3,6],[-5,3]],
 sor ),        \n         plot( [ [-6,0],[-1,0],[-8,2],[-6,0]], sgn ) \+
);" }}}{PARA 0 "" 0 "" {TEXT -1 29 "\n\nSqaures and rectangles ...." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "display( plot( [ [3,0],[7,
0],[7,6],[3,6],[3,0]], sgy ),\n         plot( [ [-5,1],[5,1],[5,3],[-5
,3],[-5,1]], sgn ),\n         plot( [ [2,2],[4,2],[4,4],[2,4],[2,2]], \+
sor ),\n         plot( [ [-7,5],[-7,2],[-2,2],[-2,5],[-7,5]], sbe ) \n
          );" }}}{PARA 0 "" 0 "" {TEXT -1 21 "\n But there are more!" 
}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 263 19 "2. Regular Po
lygons" }}{PARA 0 "" 0 "" {TEXT -1 150 "\nRegular polygons are polygon
s where all the sides are equal and all the interior angles are equal.
 A five sided regular polygon is called a pentagon." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 70 "RegularPolygon( RegPoly,5,o,1):\ndisplay( d
raw(RegPoly, scanft, cbl ));" }}}{PARA 0 "" 0 "" {TEXT -1 39 "\nAll re
gular polygons fit into circles." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 61 "display( draw(UC, scant2, cbe), draw(RegPoly, scanft, cbl ));
" }}}{PARA 0 "" 0 "" {TEXT -1 40 "\nHere are the details of this penta
gon. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "detail(RegPoly);" }
}}{PARA 0 "" 0 "" {TEXT -1 108 "\nAlthough it requires some translatio
n, what it is saying is that the interior angles are this many degrees
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "(3/5)*Pi*(180/Pi);" }}}
{PARA 0 "" 0 "" {TEXT -1 58 "\n\nHere are some other regular polygons.
 Can you name them?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Regu
larPolygon( RegPoly ,3,o,1):\ndisplay( draw(UC, scant2, cbe), draw(Reg
Poly, scanft, cgy ));\ndetail(RegPoly);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 111 "RegularPolygon( RegPoly ,6,o,1):\ndisplay( draw(UC, \+
scant2, cbe), draw(RegPoly, scanft, cgn ));\ndetail(RegPoly);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "RegularPolygon( RegPoly ,8,
o,1):\ndisplay( draw(UC, scant2, color=black), draw(RegPoly, scanft, c
or ));\ndetail(RegPoly);" }}}{PARA 0 "" 0 "" {TEXT -1 252 "\nHere is a
 regular 17-sided polygon, which has some historical significance (aft
er find the first construction of such a polygon, a young Gauss decide
d on a very fruitful life in mathematics). Notice how close it is to t
he circle that circumscribes it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 112 "RegularPolygon( RegPoly ,17,o,1):\ndisplay( draw(UC, scant2, \+
cbe), draw(RegPoly, scanft, cbl ));\ndetail(RegPoly);" }}}{PARA 0 "" 
0 "" {TEXT -1 209 "\n\n\nIf we draw a number of regular polygons on th
e same diagram we can see the progression. Here are regular polygons w
ith 3 to 10 sides, and then 3 to 30 sides. They seem to fill the unit \+
circle more and more." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Reg
PolyPlot(6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "RegPolyPlot
(10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "RegPolyPlot(25);" 
}}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 1 " " }{TEXT 266 30 "3. Interior Angles of Polygons" }}{PARA 0 "" 
0 "" {TEXT -1 124 "\nWe know that the sum of the angles of a triangle \+
is 180 degrees, and rectangle (or any four sided polygon) is 360 degre
es.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "n := 3; total_int_a
ngles := 180;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "n := 4; to
tal_int_angles := 360;" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\nActually the
re is a formula :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "total_i
nterior := n -> 180*(n-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
135 "A := array( [seq( [ n, total_interior(n) ],   n = 2..10) ]):\nA[1
,1] := `Number of sides`: \nA[1,2] := `Total Interior Angles`:\nprint(
A);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 1 " " }{TEXT 264 32 "4. Perimeter of Regular Polygons" }}
{PARA 0 "" 0 "" {TEXT -1 113 "\nIf we there is a regular polygon with \+
10 sides, and each side is 3 feet long, the perimeter is easy to compu
te.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "10*3;" }}}{PARA 0 "
" 0 "" {TEXT -1 282 "\n On the other hand, if we have a regular polygo
n with 7 sides, and the distance from the center to any of the vertice
s is 5 feet, how long is the perimeter? This is a problem that can not
 be answered adequately until you take trigonometry, but we can use Ma
ple to find the answer.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 
"RegularPolygon( RegPoly ,7,o,5):\ndisplay( draw(RegPoly, scanft, cgy \+
));\ndetail(RegPoly);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "P \+
= evalf( 70*sin(Pi/7), 20);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 265 27 "5. Area of Regular Polygons" }}{PARA 0 "" 0 "" {TEXT 
-1 70 "\nA regular polygon with n sides can be cut into n isocoles tri
angles.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "RegularPolygon
( RegPoly ,7,o,1):\npoint(A,[cos(2*Pi/7), sin(2*Pi/7)]);\npoint(B,[cos
(4*Pi/7), sin(4*Pi/7)]);\npoint(C,[cos(3*Pi/7)*cos(Pi/7), sin(3*Pi/7)*
cos(Pi/7)]);\nsegment( rad1, [o, A]); \nsegment( rad2, [o, B]); \nsegm
ent( h, [o, C]);\nsegment( side, [A,B]); \n \n\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 311 "display( draw( \n          [ UC(thickness=1,
cbe), RegPoly(thickness=2,cgn),\n            rad1(color=red,thickness \+
= 2), \n            rad2(color=gold, linestyle = 3),\n            h(th
ickness = 2, color = coral), \n            side(thickness = 3, color =
 green)\n          ]), axes=none, scaling=constrained        );" }}}
{PARA 0 "" 0 "" {TEXT -1 277 "\nYou need three pieces of information :
\n        1. n, the number of sides\n        2. the height of each tri
angle (the solid yellow/orange line in the middle of the triangle)\n  \+
      3. the length of one side of the polygon.\n\nWe can get this fro
m the detail command ....\n     \n " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "detail(RegPoly);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "Area = 7*sin(Pi/7)*cos(Pi/7): evalf(%, 50);" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 8 "6. Stars" }}
{PARA 0 "" 0 "" {TEXT -1 182 "\nAnother type of polygon, which is not \+
\"regular\", but is symetrical, is a \"star\" - a shape not usually co
vered in geometry books, but certainly common in artwork and national \+
flags.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "RegularStarPoly
gon( star,5/2,o,2):\ndraw(star,filled = true, axes = none,scaling=cons
trained,color=orange);" }}}{PARA 0 "" 0 "" {TEXT -1 253 "\nNotice that
  a star fits nicely within a regular polygon. A 5 sided star fits int
o a a regular 5 sided polygon. Also, if you cut off the five points as
 the vertices closest to the center, you will see that there is anothe
r pentagon inside the star too!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 168 "RegularPolygon( P2,5,o,2):\nRegularStarPolygon( star,5/2,o,2)
:\ndraw( [P2(color=coral), star(color = red, filled = true)], \n      \+
    axes = none, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 
45 "\nOf course stars can come in other sizes too." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 122 "RegularStarPolygon( star,9/2,o,1):\ndraw(st
ar, \n         filled = true, axes = none, scaling = constrained, colo
r=sienna );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n         \+
\251 2002 Waterloo Maple Inc " }}}{MARK "0 1" 26 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }