Concentric Circles

Concentric circles are defined as two or more circles that share the same center point, known as the midpoint, but each has a different radius. If circles overlap yet have different centers, they do not qualify as concentric circles. According to Euclidean Geometry, two concentric circles must have two different radii. The space between the circumference of these two circles is called the annulus of a circle.

In this article, we will learn about concentric circles, the theorem on concentric circles, the region between the concentric circles, its equation, and examples in detail.

What are Concentric Circles?

Concentric circles are a collection of circular shapes positioned such that they all share the same central point but have varying sizes as determined by their respective radii. These circles are akin to a set of ripples expanding outward from a singular source, or like a series of nested circular boundaries within one another.

They maintain a symmetrical arrangement around a common center, resembling a target board or a bullseye pattern, and are frequently employed in various fields, including mathematics, engineering, and design, to depict proportional relationships or as a visual aid in illustrating concepts like depth, layers, or spatial hierarchy.

Read More: Circles in Maths

Concentric Circles Meaning

Concentric circles are circles that share the same center but have different diameters or radii. Picture multiple circles, one inside the other, like a target board. They all have the same midpoint, but their sizes vary. These circles don’t touch each other; they’re just nested within one another. The term “concentric” essentially means having a common center.

Concentric Circles Examples

When considering two concentric circles, their defining feature is the sharing of a common center while having different sizes. Imagine a scenario where one circle has a radius of 6 centimeters and another circle is positioned within it with a smaller radius of 3 centimeters.

These circles are concentric because they maintain the same midpoint despite the variation in their radii. Picture this as a target in archery: the bullseye (smaller circle) sits perfectly aligned within the larger circle, both having the same center point. This arrangement makes it easy to discern the shared center point while observing the differences in the circles’ sizes.

Region between Two Concentric Circles

Think about two circles. They’re like round shapes, one inside the other sharing the same center but with different sizes. Now, the space between the edges of these circles kind of like the space between two hula hoops is what we call the “region between two concentric circles”.

Imagine these circles like a target—a smaller bullseye inside a larger circle. The space between these circles is like the area on the target board between the center and the outer ring. It’s this space that’s interesting to explore when we’re working with shapes and measurements related to circles.

The area between two concentric circles is the region lying between their circumferences. It’s akin to a circular belt around the smaller circle extending up to the larger circle’s circumference.

An Annulus is the region between two concentric circles. To find the area A annulus of this region, subtract the area of the smaller circle from the area of the larger circle and the formula for the same is,

Area of Annulus = π(r₂² − r₁²)

Concentric Circle Theorem

Concentric Circle Theorem states that, “If the chord of outer circle touches the inner circle at one point, the chord is bisected at the point of contact.”

Proof:

Consider two concentric circles with center O and radii r (larger circle) and R (smaller circle).

Draw a chord AB of the larger circle that touches the smaller circle at point C. Mark the center of the circles as O.

Now, connect OC, which is the radius of the smaller circle.

In △OAC and △OBC:

OA = OB (Both are radii of the larger circle)

OC = OC (Common side, the radius of the smaller circle)

∠OAC = ∠OBC = 90° (Perpendicular to the tangent at the point of contact is 90 degrees)

So, △OAC and △OBC are congruent by the hypotenuse-leg congruence criterion.

By the congruence of triangles, AC = BC. Therefore, the chord AB is bisected at point C, where it touches the smaller circle.

This phenomenon occurs because the tangent to a circle is perpendicular to the radius at the point of contact. Hence, the two segments of the chord from the point of contact to the ends of the chord are equal in length.

Equation of Concentric Circles

The equation of concentric circles can be represented as follows:

• For larger circle with radius R and center at the origin (0,0): x² + y² = R²
• For smaller concentric circle with radius r: x² + y² = r²

In these equations, the (x, y) coordinates on the plane satisfy the respective circle’s equation defining the points on the circle’s circumference.

Read More:

• Area Of Circle
• Circumference Of Circle

Solved Examples on Concentric Circles

Example 1: Two concentric circles have radii of 5 cm and 3 cm. Calculate the area between the two circles?

Solution:

Area between two concentric circles is the difference in the areas of the larger circle and the smaller circle.

Given:

• Radius of the larger circle, R₁ =5 cm
• Radius of the smaller circle, R₂ = 3 cm

Area of the larger circle is A₁ = R₁² = π5² = 25π cm²

Area of the smaller circle is A₂ = R₂² = π3² = 9π cm²

Area between the two circles = A₁ − A₂ = 25π − 9π= 16π cm²

Therefore, the area between the two concentric circles is 16π cm²

Example 2: Draw two concentric circles with radii 2 cm and 5 cm?

Solution:

Concentric circle with radius 2 cm and 5 cm is,

People Also Read:

• Equation of a Circle
• Radius of the circle
• Sector Of A Circle
• Tangent Of A Circle

Practice Questions on Concentric Circles

Question 1: Two concentric circles have radii 20 cm and 25 cm. Find the perimeter of the region between these circles?

Question 2: Given the larger circle’s radius as 18 units and the area between the circles as 200π sq. units, what is the radius of the smaller circle?

Question 3: Two concentric circles have radii 40 cm and 50 cm. Find the perimeter of the region between these circles?

Question 4: If the radii of two concentric circles are 15 cm and 8 cm respectively, determine the difference between the areas of the two circles?

Question 5: If the radii of two concentric circles are 20 cm and 22 cm respectively, determine the difference between the areas of the two circles?

FAQs on Concentric Circles

What is the Difference Between Concentric and Eccentric Circles?

Concentric circles share the same center, while eccentric circles have different centers.

What is an Annulus?

An Annulus is the region between two concentric circles.

How to Find the Area of the Annulus?

Subtract the area of the smaller circle from the area of the larger circle,

Annulus Area = π(r₂² −r₁²)

What are Concentric Circles?

Concentric circles are circles that share the same center point but have different radii.

How to Find Area Between Concentric Circles?

Subtract the area of the smaller circle from the larger circle’s area.

How do Radius of Circles Affect Area Between Them?

The larger the difference between the radii, the greater the area between the concentric circles.

Are Concentric Circles Congruent?

No, concentric circles have different radii making them distinct circles despite sharing a center point.

How are Concentric Circles Used in Art?

In the world of art, concentric circles are often employed to convey movement, rhythm and balance. Artists utilize this geometric pattern to create visually dynamic compositions.

Are Concentric Circles Applied in Architecture?

Yes, concentric circles find application in architecture where they symbolize unity. This geometric arrangement is incorporated into structures to create a sense of balance contributing to the overall design aesthetics.

What is Concentric Ring?

Concentric rings are a series of rings, where each ring have smaller radius inside the circle.