"Concentric" means to have a same centre. Concentric figures or objects share a common centre, axis or origin with one inside the other. Concentric circles are circles of different sizes which have a common centre. In other words we can say that, two or more circles which share the same centre are called concentric circles. Concentric circles have same centre but different radii, for it the radius is same they become equal. Since the concentric circles have different radii and same centre, they are at the same distance apart all the way around and they fit inside each other.

The picture above shows the concentric circles (5 circles) with centre “C”.

The best example for concentric circles is the evenly spaced circles of a target used in target archery or firearms. The centre of these concentric circles is called the “bulls eye” in target archery or the dart game.

## Area of Concentric Circles

To find the area of circle we use the formula πr2, where π is the mathematical constant which is approximately equal to 3.142 and “r” is the radius of the circle. Concentric circles have two or more circles involved. So this formula can be used even for concentric circles. The area between any two concentric circles is called the Annulus which is like a ring.

The area of this ring or annulus can be calculated by subtracting the area of the smaller circle from the area of the bigger circle.

In this figure above we have two concentric circles each of radius “r” and “R”.

The area of each circle is given by πr2 and πR2.

Thus the area of Annulus is given by πR2 - πr2 = π (R2 – r2)

## Non Concentric Circles

"Non-concentric" means do not to have a same centre. Non concentric circles are circles of different sizes which do not have a common centre. In other words we can say that, two or more circles which do not share the same centre are called Non concentric circles. Two or more circles which are one inside the other but not with the same centre are called non concentric circles.

## Concentric Circles in Nature

There are many places where we find concentric circles in nature. Some of them are:
• The rings in the bark of a tree.
• The circles that make the pupil and the iris in the eye.
• The ripples that are formed when a stone is tossed into water.
• The board used for archery targets
• The dart board with bulls eye
• A wheel with a hubcap.
• Patterns on Butterfly and Peacock wings.
• Onion when cut into two halves.

## Examples of Concentric Circles

### Solved Examples

Question 1: Find the area of concentric circles with two circles of radius 3cm and 4cm each.
Solution:

The radius of smaller circle, r = 3cms

The radius of the larger circle ,R = 4cms.

Area of the concentric circles = π (R2 – r2).

= π ( 42 – 32)

= π (16 -9) = 3.142 $\times$ 7 = 21.99.

Area of given concentric circles is 21.99 sq cms.

Question 2: Two concentric circles have radii of 10 and 6 respectively. Find the length of a chord of the larger circle that is tangent to the smaller circle.
Solution:

Draw two circles with common centre O, radii 10 and 6 as shown in the picture below.

Draw a tangent to the inner circle at T and extend the tangent to the outer circle at A and B.

Draw AO and BO consider triangle ABO.

By tangency,

TO = 6 (radius of inner circle) BO = 10 (radius of outer circle)

Therefore, TB = $\sqrt{(10^2 - 6^2)}$ = 8.

Similarly, we get AT = 8.

Thus, AB = AT + TB = 8 + 8 = 16.

So, the length of a chord of the larger circle with radius 10 cm that is tangent to the smaller circle with radius 6 cm is 16 cm when the two circles are concentric.