Polygons in polygons

An n-sided regular polygon is inscribed in a similar regular polygon. You can choose the number of sides [from 3 to 12] using the slider. You can vary the inscribed polygon by dragging the GOLD dot. As a function of the number of sides and the position of the GOLD dot - - what fraction of the area of the Blue polygon is covered by the Green polygon? - what is the ratio of the perimeters of the two polygons? What other questions [could,would] you ask?
Adapted from an applet by Jessica Layton