Annulus Calculator

Annulus (Ring Shape)

An annulus is the flat region between two concentric circles: a larger outer circle and a smaller inner circle sharing the same center. Think of a washer, a ring donut cross-section, a CD/DVD, or a circular frame.

Formulas:

Variables:

• R = outer radius (larger circle)
• r = inner radius (smaller circle, must be less than R)

Useful alternative form: A = pi * (R + r) * (R - r) = pi * (R + r) * w

This shows that annulus area depends on both the mean radius and the ring width.

Worked example: a washer with R = 5 cm and r = 3 cm

• Area: pi * (25 - 9) = 16pi ≈ 50.27 cm^2
• Outer perimeter: 2pi × 5 ≈ 31.42 cm
• Inner perimeter: 2pi × 3 ≈ 18.85 cm
• Total perimeter: 2pi × 8 ≈ 50.27 cm
• Ring width: 5 - 3 = 2 cm

The half-area radius, which is not where you would guess

Ask someone to draw a circle inside a disc so that the inner disc and the surrounding ring have equal areas, and almost everyone draws it at half the radius. That gives the inner disc a quarter of the area, not half, because area goes with the square of the radius.

The correct answer is R/√2, about 0.707R. It is the reason a dartboard’s rings look uneven when you measure them, and the reason the middle ring of a target has to be so much thinner than the bullseye to be worth the same score.

Real-world annuli: Pipe cross-sections, washers, circular frames, road roundabouts, and the rings of Saturn are all annuli. So is the readable part of a CD, between the clamping hub and the outer edge.

The shape also turns up in probability. Scatter points uniformly across a disc and ask where they land: the chance of falling between radius r and R is the annulus area over the disc area. Points cluster toward the rim, not the centre, because there is simply more room out there. This is the same effect that makes randomly thrown darts miss the bullseye far more often than beginners expect.