Three circles each of radius 3.5 cm touch one another. The area subtended between them is
Calculating the Area Between Three Touching Circles
This problem asks us to find the area of the region enclosed when three circles of equal radius touch each other externally. Let's break down the geometry involved and use the given radius to find the required area.
When three circles of the same radius touch each other, their centers form a special type of triangle. Let the radius of each circle be \(r\). In this problem, the radius \(r = 3.5\) cm.
Geometry of Touching Circles
Consider the centers of the three circles, let's call them \(C_1\), \(C_2\), and \(C_3\). Since the circles touch each other externally, the distance between the centers of any two touching circles is equal to the sum of their radii. As all circles have the same radius \(r\), the distance between \(C_1\) and \(C_2\) is \(r+r=2r\). Similarly, the distance between \(C_2\) and \(C_3\) is \(2r\), and the distance between \(C_3\) and \(C_1\) is \(2r\).
This means the triangle formed by connecting the centers \(C_1C_2C_3\) is an equilateral triangle with side length \(a = 2r\).
Given \(r = 3.5\) cm, the side length of the equilateral triangle is \(a = 2 \times 3.5 = 7\) cm.
Finding the Area of the Equilateral Triangle
The area of an equilateral triangle with side length \(a\) is given by the formula:
\(\text{Area of Triangle} = \frac{\sqrt{3}}{4} a^2\)
Substituting \(a = 7\) cm:
\(\text{Area of Triangle} = \frac{\sqrt{3}}{4} (7)^2 = \frac{49\sqrt{3}}{4}\) square cm.
Identifying and Calculating the Area of the Sectors
The area subtended between the three circles is the region inside the equilateral triangle \(C_1C_2C_3\) that is not covered by the circles. Inside this triangle, each circle cuts out a sector.
Since \(C_1C_2C_3\) is an equilateral triangle, each interior angle is \(60^\circ\). Thus, the angle of the sector from each circle within the triangle is \(60^\circ\).
The area of a sector of a circle with radius \(r\) and angle \(\theta\) (in degrees) is given by:
\(\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2\)
For each sector in our problem, \(r = 3.5\) cm (\(\frac{7}{2}\) cm) and \(\theta = 60^\circ\).
\(\text{Area of one Sector} = \frac{60^\circ}{360^\circ} \times \pi (3.5)^2 = \frac{1}{6} \times \pi \left(\frac{7}{2}\right)^2 = \frac{1}{6} \times \pi \frac{49}{4} = \frac{49\pi}{24}\) square cm.
There are three such sectors, one originating from the center of each circle, within the triangle. The total area covered by these three sectors inside the triangle is:
\(\text{Total Area of Sectors} = 3 \times \text{Area of one Sector} = 3 \times \frac{49\pi}{24} = \frac{147\pi}{24} = \frac{49\pi}{8}\) square cm.
Calculating the Area Subtended Between the Circles
The area subtended between the three circles is the area of the equilateral triangle minus the total area of the three sectors located within the triangle.
\(\text{Area between Circles} = \text{Area of Triangle} - \text{Total Area of Sectors}\)
\(\text{Area between Circles} = \frac{49\sqrt{3}}{4} - \frac{49\pi}{8}\)
To simplify, find a common denominator, which is 8:
\(\text{Area between Circles} = \frac{2 \times 49\sqrt{3}}{8} - \frac{49\pi}{8} = \frac{98\sqrt{3}}{8} - \frac{49\pi}{8}\)
Factor out the common term \(\frac{49}{8}\):
\(\text{Area between Circles} = \frac{49}{8} (2\sqrt{3} - \pi)\) square units.
Result Summary
| Description | Value |
|---|---|
| Radius of each circle (\(r\)) | 3.5 cm or \(\frac{7}{2}\) cm |
| Side length of equilateral triangle (\(a\)) | 7 cm |
| Area of equilateral triangle | \(\frac{49\sqrt{3}}{4}\) sq cm |
| Angle of each sector within triangle | \(60^\circ\) |
| Area of one sector | \(\frac{49\pi}{24}\) sq cm |
| Total area of three sectors | \(\frac{49\pi}{8}\) sq cm |
| Area between circles | \(\frac{49}{8} (2\sqrt{3} - \pi)\) sq units |
The calculated area subtended between the three circles is \(\frac{49}{8} (2\sqrt{3} - \pi)\) square units.
Revision Table: Key Concepts
| Concept | Formula/Definition | Relevance to Problem |
|---|---|---|
| Touching Circles | Distance between centers is sum of radii. | Forms an equilateral triangle of centers. |
| Equilateral Triangle | All sides equal, all angles \(60^\circ\). Area = \(\frac{\sqrt{3}}{4} a^2\). | Formed by centers; its area is starting point. |
| Circle Sector | Portion of circle defined by central angle. Area = \(\frac{\theta}{360}\pi r^2\). | Area within triangle subtracted from triangle area. |
| Area Subtended | Region enclosed by boundaries, not covered by shapes within. | The final area we need to calculate. |
Additional Information: Related Geometry Problems
Understanding the area between touching circles is a common geometry problem. Similar problems can involve:
- Finding the area of a region enclosed by more circles (e.g., four circles in a square).
- Calculating the area of the largest circle that can fit in the space between touching circles.
- Problems involving circles packed within other shapes like squares or triangles.
These problems often require combining knowledge of basic shapes like triangles and squares with properties of circles and sectors. Always start by drawing a diagram and identifying the key geometric figures formed by the problem setup.
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