star polygons

the 'youtube' clip on doodling with stars makes an excellent introduction to this topic

the geogebra applet is helpful for checking results

there are several varieties of star polygon on Commons.Wikimedia if larger versions are needed

several other applets are helpful, including this one
or this one

if you mark off 'n' points equally around a circle and then join a point to the one 2 along you get the (n , 2) family of star polygons

and joining 3 points along (as in the diagram) gives the (n, 3) star polygons

students who have studied exterior and interior angles of regular polygons can be asked to find the angles at the 'pointy' bits of the stars

some might then work towards and possibly derive a generalisation (using the rule for the external angle of any regular polygon) for each pointy angle of a regular (n , 2) star polygon is :

then, if we still used LOGO in classrooms, students could be asked to draw the nested star polygons, using sub-routines:

for the (n , 3) family of star polygons:

the task is a little more complex, probably involving angles in kites (or symmetrical arrowheads)

the generalisation for the pointy angles of this family (n , 3) can involve the exterior and interior angle of a regular polygon and is closely related to and can involve the (n , 2) rule:

what happens for other star polygons?

the (n , 4) family:

as might be anticipated, has a generalisation

successive generalisations can be developed either from an appreciation that (n , p) stars have (n , p - 1) stars within them:       

the (12 , 5) star has a (12 , 4) star within it

and the (12 , 4) star has a (12 , 3) star within it etc....












or from using one of the circle theorems

or by considering the numbers of whole turns when you go around a star polygon: