Calculating String Length Around Touching Circles

Let's break down how to find the length of a string tightly tied around three circles of equal radius that are touching each other. This is a common geometry problem involving tangents and arcs.

Understanding the Setup

We have three circles, each with a radius of 7 cm. When three identical circles touch each other, their centers form the vertices of an equilateral triangle. The distance between the centers of any two touching circles is the sum of their radii. Since the radii are equal (7 cm), the distance between the centers of any two touching circles is \(7 \text{ cm} + 7 \text{ cm} = 14 \text{ cm}\).

Therefore, the centers of the three circles form an equilateral triangle with a side length of 14 cm.

Analyzing the String Path

The string wrapped tightly around the circles will consist of two types of segments:

1. Straight segments that are tangent to two adjacent circles.
2. Curved segments that follow the circumference of each circle.

Calculating the Length of Straight Segments

Consider the string segment between two touching circles. This segment is a common external tangent to the two circles. If we draw radii to the points of tangency, these radii are perpendicular to the tangent string segment. Connecting the centers of these two circles forms a rectangle with the radii and the tangent segment (or a square, in this case, if we consider the segment parallel to the line connecting centers). The length of this straight tangent segment is equal to the distance between the centers of the two circles.

Since the centers form an equilateral triangle with side length 14 cm, there are three such straight segments, and each has a length equal to the side length of the triangle formed by the centers.

• Length of one straight segment = Distance between centers = \(14 \text{ cm}\).
• Total length of the three straight segments = \(3 \times 14 \text{ cm} = 42 \text{ cm}\).

Calculating the Length of Curved Segments

Now let's look at the curved parts of the string. These parts follow the circumference of each circle. At each corner where the string transitions from a straight segment to a curved segment around a circle, the string turns. Consider the angle turned around one circle.

The internal angle of the equilateral triangle formed by the centers is \(60^\circ\). At the point where the string touches a circle, the radius is perpendicular to the straight tangent segment. If we consider the center of one circle and the two points where the string touches it, the angle formed at the center by the two radii to these tangent points can be determined.

Around the center of each circle, we have the internal angle of the equilateral triangle (\(60^\circ\)) and two right angles (\(90^\circ\) each) formed by the radii and the tangent segments. The remaining angle at the center is the angle subtended by the curved part of the string on that circle. This angle is \(360^\circ - 90^\circ - 90^\circ - 60^\circ = 120^\circ\). Alternatively, think about the total turn: the string makes a \(360^\circ\) turn in total as it goes around the three circles. Since the three corners are identical, the turn at each corner is \(360^\circ / 3 = 120^\circ\).

So, the string follows an arc corresponding to a \(120^\circ\) sector on each of the three circles. The total angle covered by the curved segments is \(3 \times 120^\circ = 360^\circ\).

An arc with a total angle of \(360^\circ\) on circles of radius 7 cm is equivalent to the circumference of one circle with radius 7 cm.

• Radius of the circle = \(7 \text{ cm}\).
• Circumference of the circle = \(2 \pi r = 2 \pi (7 \text{ cm}) = 14\pi \text{ cm}\).
• Total length of the curved segments = \(14\pi \text{ cm}\).

Total Length of the String

The total length of the string is the sum of the lengths of the straight segments and the curved segments.

Total length = Total straight length + Total curved length

Total length = \(42 \text{ cm} + 14\pi \text{ cm}\)

The length of the string is \(42 + 14\pi\) cm.

Revision Table: String Around Touching Circles

Additional Information: String Around Multiple Touching Circles

This problem is a specific case of finding the length of a belt or string around multiple touching circles of the same radius.

• General Formula: For \(n\) identical circles of radius \(r\) arranged symmetrically (e.g., in a regular polygon shape for their centers), the total length of the string tightly wrapped around them is given by:
  Total Length = \(n \times (\text{distance between centers of adjacent circles}) + \text{Circumference of one circle}\)
• For \(n\) circles in a line: If \(n\) circles of radius \(r\) are placed in a line touching each other, the centers form a line. The string would wrap around the two end circles and form straight segments between adjacent circles. The total length would be \(2 \times (\text{distance along the line of centers}) + \text{Circumference of one circle}\). The distance along the line of centers is \((n-1) \times 2r\). The curved parts would be semi-circles at each end. Total length = \(2 \times (n-1) \times 2r + 2 \times \pi r = 4(n-1)r + 2\pi r\).
• For \(n\) circles forming a regular polygon of centers: Like our case with \(n=3\) (equilateral triangle), the straight parts are equal to the side length connecting centers (\(2r\)), and the total curved parts always add up to the circumference of one circle (\(2\pi r\)). The number of straight parts is \(n\). So, the total length is \(n \times 2r + 2\pi r = 2nr + 2\pi r = 2r(n+\pi)\). For \(n=3\) and \(r=7\), this gives \(2 \times 7 \times (3+\pi) = 14(3+\pi) = 42 + 14\pi\), matching our result.

This general formula \(2r(n+\pi)\) is very useful for similar problems with different numbers of touching circles forming a regular polygon shape.