consider a set of nonintersecting circles of radius r with centers at the vertices of a convex n-sided polygon having sides of lengths d1, d2,.., dn. How lon is the belt that fits around these circles 

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

Consider a set of nonintersecting circles of radius r with centers at the vertices of a convex n-sided polygon having sides of lengths d1, d2,.., dn. How long is the belt that fits around these circles?

Explore how to calculate the belt length that fits around nonintersecting circles at the vertices of a convex n-sided polygon. This problem involves geometric concepts like straight segments, circular arcs, and the sum of internal angles of polygons. Suitable for students in Grades 10-12.

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