Three circles of radius 6 cm are kept touching each other. The string is tightly tied around these three circles. What is the length of the string?
36 + 12π cm
Understanding the String Length Around Three Touching Circles
The problem asks for the length of a string tied tightly around three circles of the same radius that are touching each other. Let the radius of each circle be \(R\).
When three circles of equal radius touch each other, their centers form an equilateral triangle. The distance between the centers of any two touching circles is the sum of their radii, which is \(R + R = 2R\). Therefore, the side length of the equilateral triangle formed by the centers is \(2R\).
In this problem, the radius is given as 6 cm. So, \(R = 6\) cm. The side length of the equilateral triangle formed by the centers is \(2 \times 6 = 12\) cm.
Components of the String Length
The string length consists of two parts:
- Straight sections that are tangential to the circles.
- Curved sections that follow the circumference of the circles.
Calculating the Length of Straight Sections
There are three straight sections of the string, each connecting two adjacent circles tangentially. Due to the symmetry of the arrangement, the length of each straight section is equal to the distance between the centers of the two circles it is tangent to, which is \(2R\).
Length of each straight section = \(2R = 2 \times 6 = 12\) cm.
Total length of the three straight sections = \(3 \times (2R) = 3 \times 12 = 36\) cm.
Calculating the Length of Curved Sections
Now, let's consider the curved sections of the string. These are arcs of the circles.
At the center of each circle, radii drawn to the points where the string touches the circle (points of tangency) are perpendicular to the straight sections of the string. The angle inside the equilateral triangle at each vertex (which is a circle's center) is \(60^\circ\).
Consider one circle's center. The angles around this center are formed by the two radii to the points of tangency and the lines connecting the center to the adjacent centers (which form the equilateral triangle). The angles between the radii and the straight tangential sections are \(90^\circ\) each. The angle inside the equilateral triangle is \(60^\circ\). The total angle around the center is \(360^\circ\).
The angle corresponding to the curved section of the string around this center is the remaining angle:
Angle of curved section = \(360^\circ - 90^\circ - 90^\circ - 60^\circ = 120^\circ\).
There are three such curved sections, one around each circle.
Total angle covered by the three curved sections = \(3 \times 120^\circ = 360^\circ\).
An arc that covers a total angle of \(360^\circ\) of a circle is equal to the full circumference of the circle.
Length of the curved sections = Circumference of one circle = \(2\pi R = 2\pi \times 6 = 12\pi\) cm.
Total Length of the String
The total length of the string is the sum of the lengths of the straight sections and the curved sections.
Total string length = (Total length of straight sections) + (Total length of curved sections)
Total string length = \(36 \text{ cm} + 12\pi \text{ cm}\).
So, the length of the string is \(36 + 12\pi\) cm.
Summary of Calculation
| Component | Formula/Calculation | Length (cm) |
|---|---|---|
| Radius (R) | Given | 6 |
| Side of Center Triangle | \(2R\) | 12 |
| Length of each Straight Section | \(2R\) | 12 |
| Total Straight Length | \(3 \times 2R\) | 36 |
| Angle of each Curved Section | \(360^\circ - 2 \times 90^\circ - 60^\circ\) | \(120^\circ\) |
| Total Angle of Curved Sections | \(3 \times 120^\circ\) | \(360^\circ\) |
| Length of Curved Sections | \(2\pi R\) (for \(360^\circ\)) | \(12\pi\) |
| Total String Length | Total Straight + Total Curved | \(36 + 12\pi\) |
The calculated length of the string is \(36 + 12\pi\) cm.
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Touching Circles | When circles touch externally, the distance between their centers is the sum of their radii. |
| Equilateral Triangle | A triangle with all three sides and all three angles (\(60^\circ\)) equal. Formed by centers of three touching circles of equal radius. |
| Tangent to a Circle | A line that touches a circle at exactly one point. The radius drawn to the point of tangency is perpendicular to the tangent. |
| Arc Length | A portion of the circle's circumference. Length of an arc with angle \(\theta\) (in degrees) is \(\frac{\theta}{360^\circ} \times 2\pi R\). |
Additional Information: General Formula
For \(n\) circles of radius \(R\) arranged in a regular polygon formation and a string tied tightly around them:
- The centers form a regular n-sided polygon.
- The side length of the polygon is \(2R\).
- There are \(n\) straight sections, each of length \(2R\). Total straight length = \(n \times 2R\).
- At each center, the internal angle of the regular n-sided polygon is \(\frac{(n-2) \times 180^\circ}{n}\).
- The angle of the curved section around each center is \(360^\circ - 2 \times 90^\circ - \text{internal angle}\).
- For three circles (n=3), internal angle is \(60^\circ\). Curved angle = \(360^\circ - 180^\circ - 60^\circ = 120^\circ\). Total curved angle = \(3 \times 120^\circ = 360^\circ\). Length = \(2\pi R\). Total length = \(3 \times 2R + 2\pi R\).
- For four circles (n=4), internal angle is \(90^\circ\). Curved angle = \(360^\circ - 180^\circ - 90^\circ = 90^\circ\). Total curved angle = \(4 \times 90^\circ = 360^\circ\). Length = \(2\pi R\). Total length = \(4 \times 2R + 2\pi R\).
In general, for \(n\) circles in a regular arrangement, the total length of the string is \(n \times 2R + 2\pi R\).
Important Questions from Mensuration
The areas of three adjacent faces of a cuboidal tank are 3 m 2, 12 m 2 and 16 m 2. the capacity of the tank, in litres, is:
Volume of a cuboid is 4800 cm 3. If the height of this cuboid is 20 cm, then what will be the area of the base of cuboid ?
Two similar cubes have heights of 8 cm and 12 cm, respectively. If the capacity of the smaller cube is 80 cm 3, what is the capacity of the bigger cube (in cm 3)?
Three circles of radius 7 cm are kept touching each other. The string is tightly tied around these three circles. What is the length of the string?
A conical tent with radius 6 units and height 8 units is to be made by canvas. How much canvas is needed to make the tent? (Rounded off to two places of decimals)
