To find the length of the belt that fits around the circles with centers at the vertices of an $n$-sided convex polygon, we need to consider both the straight sides between the vertices and the arcs of the circles at each vertex.

The belt will consist of two components:

1. The straight segments between adjacent circles.
2. The arcs around each circle, since the belt wraps around the exterior.

Step-by-step breakdown:

1. Straight segments:

For each side $d_i$ of the polygon (with $i = 1, 2, \dots, n$), the belt passes tangentially to the circles. Since the circles have radius $r$, the actual distance between two tangent points along side $d_i$ will be $d_i - 2r$ (because the tangents reduce the length by the diameters of the two adjacent circles).

So, the total length of the straight segments is: $\text{Length of straight segments} = \sum_{i=1}^{n} (d_i - 2r)$

2. Arc segments:

At each vertex, the belt makes a turn around a circular arc. Since each circle has radius $r$ and there are $n$ vertices, we need to calculate the total arc length around the circles. The angle subtended by the arc at each vertex will depend on the internal angle $\theta_i$ at that vertex.

The length of the arc at each vertex is: $\text{Arc length at each vertex} = r \cdot \theta_i$ where $\theta_i$ is the angle between two sides of the polygon at the vertex. For a convex polygon, the internal angle $\theta_i$ is less than $180^\circ$.

Since the sum of the internal angles of an $n$-sided convex polygon is $(n - 2) \times 180^\circ$, the total arc length can be expressed as: $\text{Total arc length} = r \times (2\pi)$ This is because the belt wraps around the entire circumference of the circle, contributing a full $360^\circ = 2\pi$ radians in total.

Final formula for the