Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
This LOGO challenge starts by looking at 10-sided polygons then
generalises the findings to any polygon, putting particular
emphasis on external angles
Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.
Simon from Elizabeth College, Guernsey found
solutions using a spreadsheet :
We have made up a spreadsheet from Simon's description, the
columns are :
number of sides for Polygon A
exterior angle for Polygon A is 360/n
exterior angle for Polygon B is 81 - column B
number of sides required for that exterior angle is 360/column
Here's the spreadsheet file : Two Regular
Simon continues :
Polygon A starts with few sides and increases, Polygon B is
calculated for each case.
As A increases in side number, B must decrease its own side
So once A exceeds B no new solutions will be found.
For a target angle of 81 I found two solutions : 5-40 and
I then tried it with a target angle of 54
This gave me four solutions : 7-140, 8-40, 10-20 and 12-15
Finally I tried with a target angle of 27
This gave me five solutions : 14-280, 15-120, 16-80, 20-40 and
Simon then summarised the situation:
As the number of sides doubles, the angle is halved, and in
general : the number of sides is in inverse proportion to the
27 is one third of 81 so it includes solutions that are three
times the solutions for 81.
54 is double 27 so it has solutions which are half the size of
solutions for 27 provided that value is an integer.
Excellent thinking and skilful application of
a spreadsheet Simon, very well done.