Consider regular polygons with number of sides $n = 3, 4, 5, \ldots$ as shown in the figure. The center 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 center of mass for each polygon is $\Delta$. Then $\Delta$ depends on $n$ and $h$ as

Step 1: Relate the initial height to the circumradius of the polygon.

When the regular $n$-gon rests flat on one of its sides, the height $h$ of its center of mass (CM) from the ground is equal to its apothem. Let $R$ be the circumradius (the distance from the center of mass to any vertex).

$$h = R \cos\left(\frac{\pi}{n}\right)$$

Solving for $R$ gives:

$$R = \frac{h}{\cos\left(\frac{\pi}{n}\right)}$$

Step 2: Determine the maximum height of the center of mass during rotation.

As the polygon rolls about its leading vertex, this vertex acts as a fixed pivot on the ground. The distance from this pivot to the center of mass remains constant and equal to $R$. Consequently, the center of mass traces a circular arc of radius $R$.

The maximum height of the center of mass occurs when it is positioned directly above the pivot vertex.

$$h_{\text{max}} = R$$

Step 3: Calculate the maximum increase in height.

The maximum increase in height $\Delta$ is the difference between the maximum height and the initial height.

$$\Delta = R - h$$

Substitute the expression for $R$ in terms of $h$ and $n$:

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

This can also be expressed using the secant function:

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

Answer: The maximum increase in height is $\Delta = h \left( \dfrac{1}{\cos\left(\frac{\pi}{n}\right)} - 1 \right)$ or $\Delta = h \left( \sec\left(\frac{\pi}{n}\right) - 1 \right)$.