Concentric Circles: Definition, Equation, Area and Examples

As we already know that a circle is a geometric figure formed by a collection of points that are equidistant from a fixed point.The fixed point is known as the center and the equidistant distance is called the radius of the circle. In geometry, when objects share a common center, they are said to be concentric. Examples of concentric objects are circles, spheres, regular polyhedra, regular polygons, etc., as they share a common center point. According to the Euclidean Geometry, two concentric circles should have different radii from each other.

Key Terms: Concentric Circles, Annulus, Euclidean Geometry, chord, radius, center point.

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What are Concentric Circles?

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Concentric circles are two or more circles having a common center point but different radii. The region between the two concentric circles is known as the annulus.

Two concentric circles can never intersect at any point and the region and distance of the annulus between the circles will always be the same. Some examples of annulus include a ring, circular road, etc. A dartboard also has concentric circles and these circles form around a common center point called the bull’s eye.

Concentric Circles

Read more: Formation of a Differential Equation 

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Equations for Concentric Circles

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Let the equation of a circle with center (-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 will be

 x² + y² + 2gx + 2fy + c’ =0.

Similarly, a circle with the center (h, k), and the radius 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 written as:

(x – h)² + (y – k)² = r₁², where r ≠ r₁

We can obtain a family of circles by putting different values to the radius mentioned in the equation.

Thus, it can be observed that, both the equations have the same center (-g, -f), but they have different radii, where c≠ c’ .

Read more: Bayes Theorem Formula

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Concentric Circles Theorem

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Theorem - In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

Given- Two concentric circles C1 and C2, with center O and a chord AB of the larger circle C1, touching the smaller circle C2 at the point P .

To Prove- AP = BP

Proof- 

• Join OP.
• Since AB is the chord of larger circle C1, it becomes the tangent to C2 at P. And, OP is the radius of circle C2.
• Now, we know that the radius is perpendicular to the tangent at the point of contact.

Therefore, OP ⊥ AB .

• Similarly, AB is a chord of the circle C1 and OP ⊥ AB. 

Therefore, OP is the bisector of the chord AB.

• Thus, the perpendicular from the center bisects the chord, i.e., AP = BP.

Hence, PROVED.

Read more: Surface Area of a Cone Formula

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Annulus- The Region Between Concentric Circles

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The region formed between two concentric circles is called the annulus. It is shaped like a flat ring. We can calculate the area and perimeter of an annulus by using some formulas. The area of the annulus is calculated by subtracting the area of smaller circles from the area of the larger circle.

Thus, the area of Annulus can be calculated by

Area of annulus = πR² – πr²

Where, R is the radius of the larger circle and r is the radius of the smaller circle.

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Things to Remember

• For two circles to be concentric, they must have a common center point but different radii.
• Two concentric circles never intersect at any point.
• The region of the annulus, i.e., The distance between the two circles is equal all the way.
• The area of the annulus is calculated by subtracting the area of smaller circles from the area of the larger circle.
• Area of annulus = πR² – πr²