Annulus Area Calculator (Ring Area)

An annulus is the flat ring between two concentric circles. Think of a washer, a CD, the cross-section of a pipe, or the painted ring of a target. The area is the big circle minus the small circle:

A = π × (R² − r²)

Where R is the outer radius and r is the inner radius. An equivalent form, useful when you know the ring width w = R − r:

A = π × (R + r) × w

This second form is handy for race tracks and curbs, where you typically measure the width of the band directly and the mean radius.

Worked example — running track infield curb: An athletics track has an outer radius of 36.5 m and an inner radius of 35 m around each end (a 1.5 m wide curb band). The curbed-area on one curved end: A = π × (36.5² − 35²) = π × (1332.25 − 1225) = π × 107.25 ≈ 336.94 m².

If both ends of the track have the same curb, total curb material to surface that ring is roughly twice that — about 674 m².

Where annuli show up:

• Washers and gaskets. Metal washers, rubber gaskets, plumbing seals — all annular cross-sections.
• Pipe cross-section material. The metal of a pipe wall in section is an annulus. A pipe with outer radius 5 cm and inner radius 4.5 cm has wall cross-section π × (25 − 20.25) = π × 4.75 ≈ 14.92 cm².
• Race track surfacing. Asphalt or rubber surface area for the curved ends of a track.
• CD/DVD readable area. The reflective layer between the central hub and the outer edge is an annulus.
• Roundabouts and traffic circles. The drivable ring around the central island.
• Target rings. Archery and shooting targets are nested annuli.

Useful intermediate quantities:

• Width: w = R − r
• Mean radius: R_avg = (R + r) / 2
• Outer circumference: 2πR
• Inner circumference: 2πr
• Mean circumference: π × (R + r) — same as the perimeter of a circle drawn through the middle of the ring

Pro tip — checking your work:

If the ring is thin (w « R), the area is approximately the mean circumference times the width: A ≈ 2π × R_avg × w. That’s the same as unrolling the ring into a long thin rectangle. For a washer with R = 1 cm, r = 0.9 cm: exact A = π × 0.19 ≈ 0.597 cm², approximate A ≈ 2π × 0.95 × 0.1 ≈ 0.597 cm². Spot on for thin rings; diverges for fat rings.