Area of Annulus (Ring)
Learn how to calculate the area of an annulus - the ring-shaped region between two concentric circles.
Definition
- = radius of the outer circle
- = radius of the inner circle
- The result is in square units
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Worked Examples
Find the area of an annulus with outer radius cm and inner radius cm.
Identify the radii
Outer radius cm, Inner radius cm → ,
Apply the annulus formula
→
Calculate the squares
and →
Subtract and multiply by pi
→ cm
Answer: The area of the annulus is cm
Common Mistakes
Subtracting radii instead of areas:
Why it's wrong: This calculates the area of a circle with radius , not the ring between two circles.
Correct: Always subtract the areas:
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:
Forgetting that must be larger than
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 ) and which is inner (smaller )
Interactive Visual
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Practice Problems
15 problemsWhat is an annulus?
Why It Matters
- 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
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 m.
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 and :
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 mm.
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 :
Key Takeaways
- 1An annulus is the region between two concentric circles (same center, different radii)
- 2Area formula: where is outer radius and is inner radius
- 3This equals outer circle area minus inner circle area:
- 4Always identify which radius is larger before calculating
- 5Convert diameters to radii before using the formula
Frequently Asked Questions
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