Concentric circles are those circles that have a common center point. Practical examples of concentric circles include a dartboard, a wheel connected at the center by different spokes, a ripple of waves made in the pond when you throw a stone, etc.
Annulus is the region in the concentric circle between two different radii.
It resembles a flat ring.
We can find the area between the annulus as-
Let the radius of the smaller ring be r and that of the bigger ring be R.
We know the area of a circle is πr2. Therefore, here the area of the annulus will be πR2- πr2.
Example: - The diameter of two concentric circles are 14cm and 36cm away. Determine the area of the annulus.
From the formula, the area of the annulus will be πR2- πr2
Therefore, area of annulus = π362- π142
Equation of a concentric circle
Let the equation of the circle with centre (-g, -f) and radius √[g2+f2-c] be
x2 + y2 + 2gx + 2fy + c =0
Therefore, the equation of the circle concentric with the other circle be
x2 + y2 + 2gx + 2fy + c’ =0
It is observed that both the equations have the same centre (-g, -f), but they have different radii, where c≠ c’
Similarly, a circle with centre (h, k), and the radius is equal to r, then the equation becomes
(x – h)2 + ( y – k )2 = r2
Therefore, the equation of a circle concentric with the circle is
(x – h)2 + ( y – k )2 = r12
Where r ≠ r1
We will get a family of circles if we assign different values to the radius in the above equation.
Find the equation of the circle concentric with the circle x2 + y2 + 4x – 8y – 6 =0, having the radius double of its radius.
We know the equation of a circle is x2 + y2 + 2gx + 2fy + c =0
Given that x2 + y2 + 4x – 8y – 6 =0
From the given equation, the center point is (-2, 4)
Therefore, the radius of the given equation will be
r = √[g2+f2-c]
r = √[4+16+6]
r = √26
Let R be the radius of the concentric circle.
It is given that, the radius of the concentric circle is double of its radius, then
R = 2r
R = 2√26
Therefore, the equation of the concentric circle with the radius R and the center point (-g, -f ) is
( x – g )2 + ( y – f )2 = R2
(x + 2)2 + ( y – 4 )2 = (2√26 )2
x2 + 4x + 4 + y2 – 8y + 16 = 4 (26)
x2 + y2 + 4x – 8y +20 = 104
x2 + y2 + 4x – 8y – 84 = 0
Fun fact: Concentric circles play an essential role in the corporate sector. They are helpful to determine the hierarchy of different work levels and assign roles for completing a task or project.