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Annulus

An annulus is the region between two concentric circles—a flat ring or washer. Area = π(R² − r²) where R is outer radius and r is inner radius.

Concept Fundamentals
A = π(R² − r²)
Area
w = R − r
Width
2πR
Outer C
2πr
Inner C

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Washers used in 99% of bolted connections—annulus area matters for load distribution. CDs have inner radius ~23mm, outer ~58mm—the data lies in the annulus. For thin rings (R ≈ r), area ≈ 2πr(R−r) = circumference × width.

Key quantities
A = π(R² − r²)
Area
Key relation
w = R − r
Width
Key relation
2πR
Outer C
Key relation
2πr
Inner C
Key relation

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Why: Annuli appear in washers, CDs, Saturn's rings, pipe cross-sections, and archery targets. The formula subtracts inner circle from outer.

How: Area = πR² − πr² = π(R² − r²). With diameters: A = (π/4)(D² − d²). Ring width w = R − r.

Washers used in 99% of bolted connections—annulus area matters for load distribution.CDs have inner radius ~23mm, outer ~58mm—the data lies in the annulus.
Sources:Wikipedia - AnnulusWolfram MathWorld - Annulus

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Annulus CalculatorEnter outer and inner radii (or diameters) to compute ring area and width
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Annulus Calculator

Calculate the area, width, and circumferences of a ring (annulus) between two concentric circles. Step-by-step solutions with visual breakdown.

Input Dimensions

Larger circle
Smaller circle
For display
annulus_calc.out
Area
50.27 cm²
Width
2 cm
Outer C
31.42 cm
Inner C
18.85 cm
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Annulus Calculation
R = 5, r = 3
50.2655cm²
Width = 2 cmOuter C = 31.4159 cmInner C = 18.8496 cm
numbervibe.com/calculators/mathematics/geometry/annulus-calculator

Visual

Step-by-Step Breakdown

Given: Outer radius R = 5 cm, Inner radius r = 3 cm

Formula: A=π(R2−r2)A = \pi(R^2 - r^2)

Step 1: R2=52=25R^2 = 5^2 = 25

Step 2: r2=32=9r^2 = 3^2 = 9

Step 3: R2−r2=16R^2 - r^2 = 16

Step 4: A=piimes16approx50.2655;extcm2A = pi imes 16 approx 50.2655 ; ext{cm²}

Ring width: w=R−r=2;extcmw = R - r = 2 ; ext{cm}

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🧮 Fascinating Math Facts

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Annulus area = π(R² − r²) — outer circle minus inner circle.

— Formula

🔧

Washers and gaskets use annulus geometry for mechanical connections.

— Engineering

📋 Key Takeaways

  • • An annulus is the region between two concentric circles (a ring or washer)
  • • Area formula: A=π(R2−r2)A = \pi(R^2 - r^2) where R = outer radius, r = inner radius
  • • With diameters: A=π4(D2−d2)A = \frac{\pi}{4}(D^2 - d^2)
  • • Ring width: w=R−rw = R - r

Did You Know?

🔧Washers used in 99% of bolted connections
💿CDs have inner radius ~23mm, outer ~58mm
🪐Saturn's rings are annuli with R≈282,000km
🏗️Pipe cross-sections use annulus area for flow
🎯Archery target rings are concentric annuli
🍩Donuts approximate torus cross-sections (annuli)

📖 What Is an Annulus?

An annulus (plural: annuli) is the region between two concentric circles—like a flat ring or washer. It is the area of the larger circle minus the area of the smaller circle.

📝 How to Calculate Annulus Area

  1. Identify outer radius R and inner radius r (or use diameters D and d)
  2. Apply A=π(R2−r2)A = \pi(R^2 - r^2) or A=π4(D2−d2)A = \frac{\pi}{4}(D^2 - d^2)
  3. Ring width is w=R−rw = R - r

Expert Tips

Thin Ring Approximation

For a thin ring (R ≈ r), area ≈ 2πr(R−r) = circumference × width. Quick mental estimate when the ring is narrow.

Diameter vs Radius

If given diameters D and d, use A = (π/4)(D² − d²). Never mix radius and diameter in the same formula.

Units Consistency

Use the same units for R and r. Area will be in units². Convert before comparing with other shapes.

Pipe Flow Applications

Annulus area = π(R² − r²) gives the cross-sectional flow area for pipes with inner cores (e.g., heat exchangers).

📐 Formulas

The formula comes from subtracting the inner circle area from the outer: A=πR2−πr2=π(R2−r2)A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2). With diameters: A=π4(D2−d2)A = \frac{\pi}{4}(D^2 - d^2).

Rr

Annulus vs Circle vs Ellipse vs Stadium

ShapeFormulaWhen to Use
AnnulusA = π(R² − r²)Ring between two concentric circles
CircleA = πr²Single circle, no hole
EllipseA = πabOval shape, two different radii
StadiumA = πr² + 2rhRectangle with semicircular ends

Annulus by the Numbers

π
Pi constant
2
Circles (outer + inner)
R − r
Ring width
2πR + 2πr
Circumferences

🏭 Real-World Uses

  • • Washers and gaskets
  • • CDs and vinyl records
  • • Rings and donuts
  • • Pipe cross-sections

⚠️ Common Mistakes

  • • Using diameter in the radius formula (divide by 2 first, or use the diameter formula)
  • • Inner radius ≥ outer radius (annulus has zero area)

❓ Frequently Asked Questions

Can I use diameters instead of radii?

Yes. Use D=2RD = 2R and d=2rd = 2r, then A=π4(D2−d2)A = \frac{\pi}{4}(D^2 - d^2).

What if the inner radius equals the outer?

Then the annulus has zero area (no ring). The inner radius must be strictly less than the outer.

🔗 Related Calculators

Circle Area, Ellipse, Crescent Area, Torus Volume, Regular Polygon.

💡 Tip: For a thin ring (R ≈ r), the area is approximately 2πr⋅(R−r)2\pi r \cdot (R-r) (circumference × width).

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