1. What are Concentric Circles?
Concentric circles are circles that share the same centre but have different radii. They look like multiple rings drawn around a single point.
If the common centre is \(O\), and the radii are \(r_1, r_2, r_3, \ldots\), then each circle is represented by:
\(C_1 = \{P : OP = r_1\}\)
\(C_2 = \{P : OP = r_2\}\)
r_1 < r_2 < r_3 \text{ (for increasing sizes)}\)
2. Properties of Concentric Circles
Since the circles share the same centre, they have several unique geometric features.
2.1. Different Radii
The radii of concentric circles must be different; otherwise they become the same circle.
2.2. No Intersection Points
Concentric circles never intersect each other because they have the same centre but different radii.
2.3. Common Centre
The most defining feature is the same centre. All measurements are taken from this single point.
3. Distance Between Two Concentric Circles
The distance between two concentric circles is the difference of their radii.
3.1. Formula
If the radii are \(r_1\) and \(r_2\) (with \(r_2 > r_1\)), then:
\(\text{Distance} = r_2 - r_1\)
3.2. Example
If two concentric circles have radii \(5\text{ cm}\) and \(9\text{ cm}\), then the distance between them is:
9 - 5 = 4\text{ cm}\)
4. Annulus (Ring-Shaped Region)
The region between two concentric circles is called an annulus or a ring.
4.1. Area of an Annulus
If outer radius = \(R\) and inner radius = \(r\), then the area of the annulus is:
\(A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)\)
5. Real-Life Examples
Concentric circles appear in many everyday objects:
- Bullseye patterns on dartboards
- Ripples in water (waves spread as concentric circles)
- Tree trunk rings
- CDs, DVDs, and vinyl records
- Archery targets