When you connect the midpoints of adjacent sides of a regular polygon, you form a smaller regular polygon with the same number of sides. What proportion of the area of the larger polygon is the smaller polygon when the number of sides is:

i)   3?

ii)  4?

iii) 6?

iv)  n?

What is this ratio for a hectakaipentacontagon, which has 150 sides?

Thanks to Jim, Jonathan, Daniel S and Alex M-D in Y13, for their ideas for problems which together inspired this, our 150^th Weekly Maths Challenge.