The Annulus Calculator finds the area of a circular ring between an outer radius R and an inner radius r that share a center, plus ring width and both circumferences. An annulus is the region between two concentric circles. The page redraws a labelled figure with both radii marked.
Enter R and r with R greater than r. Related pages cover the full Circle Calculator for a single disk, the Tube Calculator when the ring is extruded into a pipe wall, and the Stadium Calculator for a different planar compound shape.
Calculate the area of an annulus
Ring area equals the area of the outer disk minus the inner hole when the circles share a center. Equivalently factor pi and the difference of squared radii. The Annulus Calculator requires matching length units on R and r and reports square units on the result.
A = pi * R^2 - pi * r^2 = pi * (R^2 - r^2)Area scales with the difference of squares, not with the square of the width. A thin ring far from the center can have large area because R^2 - r^2 = (R - r)(R + r) grows with the sum of the radii as well as the width.
For R = 5 and r = 3:
A = pi * (25 - 9) = 16 * pi approx 50.27Exact form 16 * pi matches the outline fixture. If the circles are not concentric, the region is not an annulus and these formulas do not apply. Eccentric rings need more advanced area methods.
Calculate the width of the ring
Width is the radial thickness of the material band measured along a line through the center. It is not the difference of diameters unless that difference is halved. The Annulus Calculator reports width beside area so thickness specs map to the radii.
width = R - rFor R = 5 and r = 3:
width = 2Manufacturing sheets that list outer diameter and wall thickness convert with R = OD/2 and r = R - width. Swapping diameter difference for radius difference doubles the width and breaks the area check against pi * (R^2 - r^2).
Calculate the inner and outer circumference
Each circumference is the perimeter of its circle at the corresponding radius. Some manufacturing checks track outer seal length and inner seal length separately on the same part. The Annulus Calculator returns both.
C_outer = 2 * pi * R
C_inner = 2 * pi * rFor R = 5 and r = 3:
C_outer = 10 * pi approx 31.42
C_inner = 6 * pi approx 18.85Mean circumference 2 * pi * ((R + r) / 2) = pi * (R + r) appears in some washer approximations for thin rings; for R = 5 and r = 3 that mean is 8 * pi. Exact area still uses the difference of squares, not mean circumference times width, though for very thin rings those estimates get close.
Solve an annulus with radii 5 and 3
Use outer radius 5 and inner radius 3 as the reference annulus. Area factorizes to 16 * pi. Width is 2. Circumferences are 10 * pi and 6 * pi. 1. Width
width = 5 - 3 = 22. Area
A = 16 * pi approx 50.273. Outer circumference
C_outer = 10 * pi approx 31.424. Inner circumference
C_inner = 6 * pi approx 18.85Extruding this annulus through length 20 produces tube material volume 320 * pi on the Tube Calculator. The shared R = 5, r = 3 fixture links the family without copying the tube page's length treatment here.
Washer and gasket problems are the everyday annulus. Measure outer and inner diameters with calipers, halve to radii, and read area for coating or stamped material. If the hole is off-center, stop: the part is not an annulus and the simple difference of disk areas no longer equals the material region.
Factoring A = pi * (R - r) * (R + r) shows why a 2-unit width at mean radius 4 (here R + r = 8, half is 4) yields 16 * pi: width times sum of radii times pi. That identity helps mental checks before trusting a calculator output.
Frequently asked questions
What is annulus area?
Annulus area is A = pi * (R^2 - r^2). For R = 5 and r = 3, A = 16 * pi approx 50.27.
What is ring width?
Width is R - r. For R = 5 and r = 3, width = 2.
What are the circumferences?
Outer circumference is 2 * pi * R; inner is 2 * pi * r. For the reference ring, 10 * pi and 6 * pi.
Must the circles be concentric?
Yes. An annulus is defined by concentric circles. Non-concentric rings are a different region.
How does an annulus relate to a tube?
Tube material volume is annulus area times length. Use the Tube Calculator when a third dimension (length) is present.
Can the Annulus Calculator use diameters?
Yes. Halve each diameter to a radius, or enter diameters if the fields convert internally before squaring.
What if the inner radius is zero?
The region becomes a full disk of area pi * R^2. Use the Circle Calculator for the single-circle case.
What units does the Annulus Calculator use?
Radii share one length unit. Area uses square units; circumferences and width use length units.
Is area equal to width times mean circumference?
For a thin ring the estimate width * 2 * pi * ((R+r)/2) is close. Exact area is always pi * (R^2 - r^2), which equals pi * (R - r) * (R + r).
What if I only know diameters?
Halve both diameters to get R and r, then apply the same formulas. Width in radius units is still R - r, which is half the difference of the diameters.
Can the Annulus Calculator solve for a missing radius?
Yes. From area and one radius, rearrange R^2 = A/pi + r^2 or r^2 = R^2 - A/pi, taking the positive root that keeps R > r.
What is the reference annulus area?
For R = 5 and r = 3, area is 16 * pi approx 50.27 square units.
How is annulus area factored?
A = pi * (R - r) * (R + r). With width 2 and R + r = 8, the product 16 * pi matches the difference of squares.
Summary
Annulus Calculator returns ring area, width and both circumferences from outer and inner radii, with substituted arithmetic and a labelled figure. Worked numbers match R = 5 and r = 3: A = 16 * pi and width 2. Use the Circle Calculator for a solid disk and the Tube Calculator when the ring becomes a pipe wall.