JUMP TO TOPIC

# Concentric Circle|Definition & Meaning

## Definition

If two **objects** in geometry have a **common** center, they are **deemed** to be **concentric.** Due to their **shared** center, regular **polygons** and **circles** are all **concentric.** In **Euclidean** geometry, two **concentric** circles always have **different radii** but the **same** center.

**The Meaning of Concentric Circles **

The circles that **share** a center are referred to as **concentric** circles, and they each have their own **unique** radius and consequently, circumference. To put it **another** way, it is described as two or even **more circles** sharing the same center point. The **space between** the** circumferences** of two concentric circles is called the **annulus**.

The **following diagram** depicts a figure with two **concentric** circles with a **common** center O.

By **selecting** the **inversion center** as one of the **restricting** points, any two **circles** could be formed **concentric** by **inversion.**

**Examples of Concentric Circles Taken from Real Life**

Have you ever **looked** at the **wheel** that’s on the ship? It consists of two **concentric** rings that are connected by spokes. **Darts** have been played by a good number of us. In addition, there are **concentric** circles on a dartboard. The circles on the **dartboard** were drawn around the bull’s eye, which serves as the standard **middle** of the line.

**What Is the Annulus?**

The **region** that is **between** two **concentric** circles is known as an **annulus.** An annulus is a **name** given to the space that is created by the intersection of two concentric circles. It has the **shape** of a ring but is **flattened** out. It is possible to **determine** the area of an **annulus** by first **determining** the size of the **outer** ring and then the **area** of the inner circle. To **acquire** the result, we need to calculate the **difference** in an area that **exists** between the two circles. The **annulus** is depicted in the **following** illustration by the **portion** that is **shaded.**

Look at **figure** 3**.** When we **subtract** the area of a **smaller circle** from that of a larger circle, we are able to calculate the area of an annulus, also known as the blue zone. The **value** of r is used to indicate the **radius** of a **smaller** circle, **whereas** R is **used** to indicate the radius of a **larger** circle. **Consequently,** the **following formula** can be **used** by us in order to **calculate** the area of an annulus:

(πR)^{2 }– (πr)^{2} = π (R^{2 }– r^{2})

## Important Points on Annulus

- The
**center**of each**concentric**circle’s**points**is always the**same.** - If two
**concentric**circles**overlap**one**another,**then the radii of those**circles**will be equal. - The
**region**enclosed by 2**concentric**circles is**known**as the**annulus.**

**Can There Be More Than Two Concentric Circles?**

It is **possible** to have more than two concentric circles. In fact, there can be** infinitely many** concentric circles around the same center point. The standard equation of a circle with radius r and center offset from the origin (a, b) :

**(x – a) ^{2} + (y – b)^{2} = r^{2}**

Consider a set of n circles c_{1}, c_{2}, …, c_{n} with **distinct radii** r_{1}, r_{2}, …, r_{n} and **equal center offsets** such that a_{1}, a_{2}, …, a_{n} = a and b_{1}, b_{2}, …, b_{n} = b. Then, these circles are **concentric** with respect to the center point **(a, b)**.

## Numerical Examples of a Concentric Circle

### Example 1

How much **area** does an **annulus** with an outside radius of **22** cm and an **inner radius** of **15** cm **have?**

### Solution

**Given that:**

The **outer radius** is 22 cm.

The **inner radius** is 15 cm.

**To find:**

The **area** of **annulus.**

Since the **area** of a circle is:

**Area = $\pi$ R$^2$ **

For the **outer circle,** the **area** is:

**Area = $\pi$ R$^2$ **

By **putting values,** we get:

**Area = 3.14 x 22 x 22**

By **calculating,** we **get:**

**=1519.76 cm$^2$**

For the **inner circle**, we know that:

**Area = $\pi$ R$^2$**

**putting values,**we get:

**Area = 3.14 x 22 x 22**

**calculating,**we

**get:**

**= 813.26 cm$^2 $**

Thus **annulus area** is:

**= 1519.76 – 813.26**

**= 706.5 cm$^2$**

### Example 2

How **much** area **does** an annulus with an outside **radius** of **52** cm and an **inner** radius of **25** cm have?

### Solution

**Given that:**

The **outer radius** is **52 cm.**

The **inner radius** is **25 cm.**

**To find:**

The **area** of **annulus. **Again, using the formula for the area of a circle:

**Area = $\pi$ R$^2$ **

For the **outer** circle, the **area** is:

**Area = $\pi$ R$^2$ **

By **putting** values, we **get:**

**Area = 3.14 x 22 x 22**

By **calculating,** we get:

**= 8490.56 cm$^2$**

**inner circle,**we

**know**that:

**Area = $\pi$ R$^2$**

**putting**values, we

**get:**

**Area = 3.14 x 25 x 25**

**calculating,**we

**get:**

**= 1962.5 cm$^2 $**

Thus **annulus area** is:

area of annulus = outer circle area – inner circle area

By **putting values,** we get:

**= 8490.56 – 1962.5**

**= 6528.06 cm$^2 $**

Thus the **area** of **annulus** for the **given** values is:

**= 6528.06 cm$^2 $**

*All images/mathematical drawings were created with GeoGebra.*