Concentric circles

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.

Solution: -

From the formula, the area of the annulus will be πR²- πr²

Therefore, area of annulus = π36²- π14²

= 864.28cm

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. 

Example 

Find the equation of the circle concentric with the circle x² + y² + 4x – 8y – 6 =0, having the radius double of its radius.

Solution 

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.