Three regular polygons, all with unit sides, share a common vertex and are all coplanar. Each polygon has a different number of sides, and each polygon shares a side with the other two; there are no gaps or overlaps. Find the number of sides for each polygon. There are multiple answers.
The three regular polygons all meet at the vertex, and if they each
share a side with both others, then they completely cover the 360
degrees around the vertex. If we write out the degrees at the
vertex for the sides of regular polygons, we have:
# sides interior angle (degrees)
3 60
4 90
5 108
6 120
8 135
9 140
10 144
12 150
15 156
18 160
20 162
24 165
infinite 180 (if allowed)
I've done this by hand (and Charlie can likely write a program to go
through them quickly), so I may have missed a few, but here are
(probably) most of the possible answers:
the angles (in degrees) shown first, the # of sides shown next:
120/120/120 6/6/6
90/135/135 4/8/8
60/150/150 3/12/12
108/108/144 5/5/10
90/120/150 4/6/12
90/108/162 4/5/20
60/140/160 3/9/18
60/144/156 3/10/15
60/135/165 3/8/24
and if we allow a degenerate "infinite-sided polygon", which is essentially a straight line:
60/120/180 3/6/infinite
90/ 90/180 4/4/infinite
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Posted by Thalamus
on 2004-07-12 12:18:48 |
solution
