Consider regular polygons with number of sides \( n=3,4 \), \( 5 . . . \) as shown in the figure. The centre of mass of all the polygons is at height \( h \) from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the centre of mass for each polygon is \( \Delta \). Then \( \Delta \) depends on \( n \) and \( h \) as:
img title=polygon.png src=https://d2bps9p1kiy4ka.cloudfront.net/5b09189f7285894d9130ccd0/2a857a95-e3b6-4145-a792-ba27e67f181c.png alt= width=245 height=73 /
(1) \( \Delta=h \sin ^{2}\left(\frac{\pi}{n}\right) \)
(2) \( \Delta=h \sin \left(\frac{2 \pi}{n}\right) \)
(3) \( \Delta=h\left(\frac{1}{\cos \left(\frac{\pi}{n}\right)}-1\right) \)
(4) \( \Delta=h \tan ^{2}\left(\frac{\pi}{2 n}\right) \)
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