Area of Annulus (Ring)

Learn how to calculate the area of an annulus - the ring-shaped region between two concentric circles.

Intermediate20 minLesson

Definition

An annulus (plural: annuli) is the region between two concentric circles - circles that share the same center but have different radii.

To find the area of an annulus:

1.Calculate the area of the outer circle (larger radius $R$ )
2.Calculate the area of the inner circle (smaller radius $r$ )
3.Subtract the inner area from the outer area

A_{\text{annulus}} = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)

Where:

$R$ = radius of the outer circle
$r$ = radius of the inner circle
The result is in square units

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What is an annulus?

Worked Examples

Find the area of an annulus with outer radius $R = 5$ cm and inner radius $r = 3$ cm.

Identify the radii

Outer radius $R = 5$ cm, Inner radius $r = 3$ cm → $R = 5$ , $r = 3$

Apply the annulus formula

$A = \pi(R^2 - r^2)$ → $A = \pi(5^2 - 3^2)$

Calculate the squares

$5^2 = 25$ and $3^2 = 9$ → $A = \pi(25 - 9)$

Subtract and multiply by pi

$A = \pi \times 16 = 16\pi$ → $A = 16\pi \approx 50.27$ cm $^2$

Answer: The area of the annulus is $16\pi \approx 50.27$ cm $^2$

Common Mistakes

Subtracting radii instead of areas: $A = \pi(R - r)^2$

Why it's wrong: This calculates the area of a circle with radius $(R - r)$ , not the ring between two circles.

Correct: Always subtract the areas: $A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)$

Confusing diameter with radius

Why it's wrong: Problems often give diameters. Using diameter instead of radius gives an area 4 times too large.

Correct: Always convert diameter to radius first: $r = d \div 2$

Forgetting that $R$ must be larger than $r$

Why it's wrong: If you accidentally swap the radii, you get a negative area, which is impossible.

Correct: Always identify which circle is outer (larger $R$ ) and which is inner (smaller $r$ )

Interactive Visual

Area and perimeter

Outer radius R6Radius r3

Area84.82

$A = \pi(R^2 - r^2) = \pi(6^2 - 3^2) \approx 84.82$

Perimeter56.55

$P = 2\pi R + 2\pi r \approx 56.55$

The ring is the big circle minus the hole: π(R² − r²).

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Practice Problems

15 problems

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What is an annulus?

Why It Matters

Annuli appear frequently in everyday life and engineering:

Pipe cross-sections: Water pipes, garden hoses, and tubes are hollow - their cross-section is an annulus
Washers and gaskets: The metal washers used with bolts are annuli
CDs and DVDs: The recordable area of a disc is an annulus (the center hole cannot store data)
Dartboards: Each scoring ring on a dartboard is an annulus
Tree rings: The annual growth rings visible in tree cross-sections

Understanding annulus area helps engineers design pipes, architects plan circular walkways, and scientists measure growth patterns.

Real World Applications

Designing a Circular Walkway

Landscape architects use annulus calculations when designing circular paths around fountains or gardens.

Example:

A circular fountain has radius 4 meters. A walkway 2 meters wide surrounds it. The walkway area is $\pi(6^2 - 4^2) = 20\pi \approx 62.83$ m $^2$ .

1Try It Yourself

A circular garden has radius 3 meters. You want to add a 1.5-meter wide path around it.

What is the area of the path?

Step 1: Write the mathematical expression

Use the annulus formula with $R = 4.5$ and $r = 3$ :

CD and DVD Storage Capacity

The recordable area of optical discs is an annulus because the center has a hole and non-recordable zone.

Example:

A standard DVD has inner recording radius 24 mm and outer recording radius 58 mm. The data area is $\pi(58^2 - 24^2) = 2788\pi \approx 8,759$ mm $^2$ .

2Try It Yourself

A mini-CD has outer recording radius 40 mm and inner recording radius 22 mm.

What is the recordable area?

Step 1: Write the mathematical expression

Calculate $\pi(40^2 - 22^2)$ :

Key Takeaways

1An annulus is the region between two concentric circles (same center, different radii)
2Area formula: $A = \pi(R^2 - r^2)$ where $R$ is outer radius and $r$ is inner radius
3This equals outer circle area minus inner circle area: $\pi R^2 - \pi r^2$
4Always identify which radius is larger before calculating
5Convert diameters to radii before using the formula

Frequently Asked Questions

Divide each diameter by 2 to get the radii. If outer diameter is

D

and inner diameter is

d

, then

A = \pi\left(\left(\frac{D}{2}\right)^2 - \left(\frac{d}{2}\right)^2\right) = \frac{\pi}{4}(D^2 - d^2)

Glossary

Annulus: A ring-shaped region between two concentric circles
Concentric circles: Circles that share the same center point but have different radii
Outer radius (R): The radius of the larger circle that forms the outside edge of the annulus
Inner radius (r): The radius of the smaller circle that forms the inside edge (hole) of the annulus