Have you observed what happens when a pebble is dropped in a pond or the ripples formed by a milk drop. They form multiple circle one after the other. Those circles are called concentric circles.
What are Concentric Circles
In geometry, objects are said to be concentric, when they share a common center. Circles, spheres, and regular polygons are all concentric as they share the common center point.
Concentric circles are defined as two or more circles that share the same center point. They fit inside each other and are of the same distance from the center.
See the diagram below. It shows 2 concentric circles having a common center point.
Concentric Circles
However, just because one circle is inside the other doesn’t necessarily mean they are concentric circles. If the given pairs of circles have a different center point, as shown below, they are not concentric.
Non Concentric Circles
When we draw two circles with the same center point, the region enclosed between them is called an annulus. It is flat-shaped like a ring.
Common Examples in Real Life
Some easy to relate examples of concentric circles in real life are shown below:
The wheel of a Ship has two concentric circles
The bulls’ eye has multiple concentric circles
A dartboard around the bullsseye
A racing track
The grooves on a vinyl record
Radius of Two Concentric Circles
If two or more circles are concentric in the same plane, they must have two different radii. Concentric circles do not ever intersect at a point and the distance between the two circles – the region of the annulus is same all the way.
Formulas
Area of an Annulus of Two Concentric Circles
The area of an annulus between two concentric circles can be determined by subtracting the area of the inner circle from the area of the outer circle
The formula to find the annulus of two concentric circles is given below:
Area of an Annulus of Two Concentric Circles
Area (A) = π(R2 – r2), here R = radius of the outer circle, r = radius of the inner circle, π = 3.141
Let us solve some examples, to clear your concept.
What is the area of the annulus formed by two concentric circles of radii 5 cm and 13 cm?
Solution:
As we know, Area (A) = π(R2 – r2), here R = 13 cm, r = 5 cm, π = 3.141 = 3.141(132 – 52) = 3.141(169 – 25) = (3.141 × 144) cm2 = 452.304 cm2
A race track is in the form of a ring. The inner radius of the field is 56 inches and the outer radius is 61 inches. Find the area of the race track.
Solution:
As we know, Area (A) = π(R2 – r2), here R = 61 cm, r = 56 cm, π = 3.141 = 3.141(612 – 562) = 3.141(3721 – 3136) = 3.141 × 585 = 1837.48 cm2 Therefore, the area of the race-track = 1837.48 cm2
Concentric Circle Equations
We know, the equation of the circle with center (-g, -f) and radius √[g2+f2-c] is
x2 + y2 + 2gx + 2fy + c =0
So, the equation of the concentric circle can be written as:
x2 + y2 + 2gx + 2fy + c’ =0
It is thus found that both the equations have the same centre (-g, -f), but they have different radii, where c≠ c’
Similarly, if a circle has a centre (h, k), and the radius ‘r’, then equation becomes
(x – h)2 + (y – k)2 = r2
Therefore, the equation of a circle concentric with the circle is written as:
(x – h)2 + (y – k)2= r12, here r ≠ r1
By assigning different values to the radius in the above equation, we can obtain a family of circles.
Let us solve an example to clear you concept even more.
Find the equation of the circle concentric with the circle x2 + y2 + 2x – 4y – 3 =0, having the radius double of its radius.
Solution:
The circle equation is given as, x2 + y2 + 2x – 4y – 3 =0 As we know, the general form of equation of a circle is given as: x2 + y2 + 2gx + 2fy + c =0 From the given equation, the center point is (-1, 2) Therefore, the radius of the concentric circle is, r = √[g2+f2-c] r = √[1+4+3] r = √8 Let, ‘R’ be the radius of the concentric circle It us given that, the radius of the concentric circle is double its radius, then R = 2r R = 2√8 Let, ‘R’ be the concentric circle with radius ‘R’ and the center point (-g, -f) is (x – g)2 + (y – f)2 = R2 (x + 1)2 + (y – 2)2 = (2√8)2 x2 + 2x + 1 + y2 – 4y + 4 = 4(8) x2 + y2+ 2x – 4y + 5 = 4(8) x2 + y2+ 2x – 4y + 5 = 32 x2 + y2+ 2x – 4y + 27 = 0
