Question 960194
how many sides does a regular polygon have if one of the exterior angles=9°? 
<pre>
The sum of the exterior angles of a polygon is 360° and since there are n
exterior angles all with the same measure,

360°/n = 9°

Multiply both sides by n

360° = 5°n

Divide both sides by 5°

72 = n

That's the answer.  

[Incidentally, a 72-sided regular polygon would be hard to tell
from a circle, unless it were very large.]

</pre>
and 

how many sides does a regular polygon have if one of the exterior angles=11.25°?
<pre>
360°/n = 11.25°

Multiply both sides by n

360° = 11.25°n

Divide both sides by 11.25°

32 = n

Answer: 32 sides. Here's how a 32-sided regular polygon looks.
The green lines are drawn from the center to the vertices, and
would all be radii of a circumscribing circle:

{{{drawing(400,400,-1.1,1.1,-1.1,1.1, 
line(1,0,0.98078528,0.19509032),
line(0.98078528,0.19509032,0.92387953,0.38268343),
line(0.92387953,0.38268343,0.83146961,0.55557023),
line(0.83146961,0.55557023,0.70710678,0.70710678),
line(0.70710678,0.70710678,0.55557023,0.83146961),
line(0.55557023,0.83146961,0.38268343,0.92387953),
line(0.38268343,0.92387953,0.19509032,0.98078528),
line(0.19509032,0.98078528,0,1),
line(0,1,-0.19509032,0.98078528),
line(-0.19509032,0.98078528,-0.38268343,0.92387953),
line(-0.38268343,0.92387953,-0.55557023,0.83146961),
line(-0.55557023,0.83146961,-0.70710678,0.70710678),
line(-0.70710678,0.70710678,-0.83146961,0.55557023),
line(-0.83146961,0.55557023,-0.92387953,0.38268343),
line(-0.92387953,0.38268343,-0.98078528,0.19509032),
line(-0.98078528,0.19509032,-1,0),
line(-1,0,-0.98078528,-0.19509032),
line(-0.98078528,-0.19509032,-0.92387953,-0.38268343),
line(-0.92387953,-0.38268343,-0.83146961,-0.55557023),
line(-0.83146961,-0.55557023,-0.70710678,-0.70710678),
line(-0.70710678,-0.70710678,-0.55557023,-0.83146961),
line(-0.55557023,-0.83146961,-0.38268343,-0.92387953),
line(-0.38268343,-0.92387953,-0.19509032,-0.98078528),
line(-0.19509032,-0.98078528,0,-1),
line(0,-1,0.19509032,-0.98078528),
line(0.19509032,-0.98078528,0.38268343,-0.92387953),
line(0.38268343,-0.92387953,0.55557023,-0.83146961),
line(0.55557023,-0.83146961,0.70710678,-0.70710678),
line(0.70710678,-0.70710678,0.83146961,-0.55557023),
line(0.83146961,-0.55557023,0.92387953,-0.38268343),
line(0.92387953,-0.38268343,0.98078528,-0.19509032),
line(0.98078528,-0.19509032,1,0),
green(
line(1,0,0,0),
line(0.98078528,0.19509032,0,0),
line(0.92387953,0.38268343,0,0),
line(0.83146961,0.55557023,0,0),
line(0.70710678,0.70710678,0,0),
line(0.55557023,0.83146961,0,0),
line(0.38268343,0.92387953,0,0),
line(0.19509032,0.98078528,0,0),
line(0,1,0,0),
line(-0.19509032,0.98078528,0,0),
line(-0.38268343,0.92387953,0,0),
line(-0.55557023,0.83146961,0,0),
line(-0.70710678,0.70710678,0,0),
line(-0.83146961,0.55557023,0,0),
line(-0.92387953,0.38268343,0,0),
line(-0.98078528,0.19509032,0,0),
line(-1,0,0,0),
line(-0.98078528,-0.19509032,0,0),
line(-0.92387953,-0.38268343,0,0),
line(-0.83146961,-0.55557023,0,0),
line(-0.70710678,-0.70710678,0,0),
line(-0.55557023,-0.83146961,0,0),
line(-0.38268343,-0.92387953,0,0),
line(-0.19509032,-0.98078528,0,0),
line(0,-1,0,0),
line(0.19509032,-0.98078528,0,0),
line(0.38268343,-0.92387953,0,0),
line(0.55557023,-0.83146961,0,0),
line(0.70710678,-0.70710678,0,0),
line(0.83146961,-0.55557023,0,0),
line(0.92387953,-0.38268343,0,0),
line(0.98078528,-0.19509032,0,0)) )}}}

Edwin</pre>