Another table, made in the same style, has its top, in the shape of an \( n \)-sided regular polygon. The length \( \mathrm{AZ}=p \mathrm{~cm} \), where Z is at the centre of the table and A is one of the corners of the table.

To analyze the situation described, we have a table with a top that is shaped like a regular \( n \)-sided polygon. The length \( \mathrm{AZ} = p \) cm indicates that point A is one of the vertices (corners) of the polygon, and point Z is the center of the polygon. ### Properties of a Regular \( n \)-Sided Polygon 1. **Vertices and Center**: A regular \( n \)-sided polygon has \( n \) vertices, all equidistant from the center. The distance from the center to any vertex is called the circumradius \( R \). 2. **Circumradius**: For a regular \( n \)-sided polygon, the circumradius \( R \) can be calculated using the formula: \[ R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} \] where \( s \) is the length of each side of the polygon. 3. **Apothem**: The apothem \( a \) of the polygon, which is the distance from the center to the midpoint of a side, can be calculated as: \[ a = R \cos\left(\frac{\pi}{n}\right) \] ### Given Information - The distance from the center \( Z \) to a vertex \( A \) is given as \( AZ = p \) cm. Therefore, we have: \[ R = p \] ### Summary In this case, the circumradius \( R \) of the regular \( n \)-sided polygon is equal to \( p \) cm. This means that the distance from the center of the table to any of its corners (vertices) is \( p \) cm. If you need further calculations or specific properties related to the polygon, such as the area or perimeter, please let me know!