Annulus Area Calculator

Calculate the area, ring width, and circumferences of an annulus (the region between two concentric circles). Enter the outer and inner radius.

Inputs

Annulus diagramTwo concentric circles with outer radius R and inner radius r, showing the annular region A and ring width w.ORrwA

Results

\begin{aligned} A &= \pi (R^2 - r^2) \\ &= \pi ((0.1\,\text{m})^2 - (0.06\,\text{m})^2) \\ &= ?\,\text{m²} \end{aligned}
\begin{aligned} w &= R - r \\ &= (0.1\,\text{m}) - (0.06\,\text{m}) \\ &= ?\,\text{m} \end{aligned}
\begin{aligned} C_R &= 2\pi R \\ &= 2\pi (0.1\,\text{m}) \\ &= ?\,\text{m} \end{aligned}
\begin{aligned} C_r &= 2\pi r \\ &= 2\pi (0.06\,\text{m}) \\ &= ?\,\text{m} \end{aligned}

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What Is an Annulus?

An annulus is the flat region between two concentric circles — the shape of a washer, a pipe cross-section, or a circular track. This calculator computes the annulus area, ring width, outer circumference, and inner circumference from the outer radius R and inner radius r.

Formulas

The area of an annulus equals the area of the outer circle minus the area of the inner circle:

A=π(R2−r2)A = \pi(R^2 - r^2)

The ring width is simply the difference between the two radii:

w=R−rw = R - r

The outer and inner circumferences follow the standard circle formula:

CR=2πR,Cr=2πrC_R = 2\pi R, \quad C_r = 2\pi r

Worked Example

A steel washer has an outer diameter of 20 mm and an inner diameter of 12 mm, giving R = 10 mm and r = 6 mm.

A=π(102−62)=π(100−36)=64π≈201.06 mm2A = \pi(10^2 - 6^2) = \pi(100 - 36) = 64\pi \approx 201.06 \text{ mm}^2

The ring width is $w = 10 - 6 = 4$ mm. This value is useful for checking whether the washer fits a bolt or a tube.

Practical Applications

Annulus calculations appear in many real-world contexts:

  • Engineering – pipe wall cross-sections, bearing rings, and flanged connections all involve annular areas for stress and flow calculations.
  • Construction – circular tracks, fountains, and ring-shaped floor tiles require annulus area for material estimation.
  • Astronomy – solar and lunar annular eclipses create a ring-shaped (annular) shadow described by this geometry.

For the full circle case (no hole), see Circle Area & Circumference Calculator.

Frequently Asked Questions (FAQ)

What is the formula for the area of an annulus?

The area of an annulus is A = π(R² − r²), where R is the outer radius and r is the inner radius. This equals the outer circle area minus the inner circle area. For example, if R = 10 cm and r = 6 cm, the area is π(100 − 36) = 64π ≈ 201.1 cm².

How do I calculate the area of a ring shape?

A ring (annulus) is the region between two concentric circles. To find its area: (1) measure the outer radius R and inner radius r, (2) apply A = π(R² − r²). If you know the diameters instead of radii, divide each by 2 first. The formula works for washers, gaskets, pipe cross-sections, and circular tracks.

What is the difference between inner radius and outer radius?

The outer radius R is the distance from the center to the outer edge; the inner radius r is the distance from the center to the inner edge (the hole). The ring width is R − r. For a solid circle with no hole, r = 0 and the formula reduces to A = πR². The outer radius must always be greater than the inner radius.

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