1. Geometry of a regular polygon rolling about a vertex

   For a regular $n$-gon resting on one side on a horizontal surface, its center of mass is at the geometric center.

   Given: the center of mass is at height $h$ from the ground.

   When the polygon starts rotating about the leading vertex (the vertex touching the ground), the center of mass moves on a circular arc centered at that vertex.

2. Distance of center from the pivot vertex

   Let $R$ be the दूरी from the center of the polygon to any vertex (circumradius).

   When one side is on the ground, the perpendicular distance of the center from that side equals the inradius: $r = R\cos\left(\frac{\pi}{n}\right)$

   But this perpendicular distance is exactly the height of the center above the ground, so $h = R\cos\left(\frac{\pi}{n}\right)$ hence $R = \frac{h}{\cos\left(\frac{\pi}{n}\right)}$

3. Maximum height during tipping

   As the polygon rotates about the leading vertex, the center rises until it comes vertically above the pivot vertex. At that instant, its height above ground is maximum and equal to the radius of rotation: $H_{\max} = R$

   Initially, the center is at height $H_{\text{initial}} = h$

   Therefore the maximum increase in height is $\Delta = R - h$

4. Substitute $R$ in terms of $h$

   Using $R = \frac{h}{\cos\left(\frac{\pi}{n}\right)}$ we get $\Delta = \frac{h}{\cos\left(\frac{\pi}{n}\right)} - h$

   $\boxed{\Delta = h\left(\frac{1}{\cos\left(\frac{\pi}{n}\right)} - 1\right)}$

5. Match with the options

   This matches Option B.

Final Answer

$\boxed{\Delta = h\left(\frac{1}{\cos\left(\frac{\pi}{n}\right)} - 1\right)}$ So, the correct option is B.