What Is an Annulus? Formula & Examples
What Is an Annulus?
So, what is an annulus, exactly? In plain terms, an annulus is the flat, ring-shaped region that remains when a smaller circle is removed from the center of a larger circle, with both circles sharing the same center point.
Think of it as a circle with a circular hole punched clean out of its middle. The word itself comes from the Latin "annulus," meaning "little ring" — the same root that gives us words like "annular" (ring-shaped) and even the biological term annelids, whose segmented bodies look like stacked rings.
An annulus is a two-dimensional, ring-shaped region bounded by two concentric circles — circles that share the same center but have different radii. The space between the inner and outer circle is the annulus itself.
What Shape Is an Annulus?
If someone asks you what shape is an annulus, the honest answer is: it's a shape of its own, distinct from both a circle and a disc. It isn't a full circle, because the middle is missing. It isn't a simple curve, because it has area and thickness. Instead, it belongs to a small, elegant family of ring shapes:
- It has two boundaries — an outer circle and an inner circle — instead of one.
- Both circles are concentric, meaning they share a single center point.
- The region between them has uniform width only when the circles are perfectly concentric; if the inner circle is off-center, the shape is technically no longer a true annulus.
- It is a two-dimensional, flat shape — the 3D version, like a car tire or a bagel, is called a torus.
You'll see the annulus shape hiding in plain sight: a CD or vinyl record, a rubber gasket, an archer target's outer bands, a lifebuoy ring, tree rings viewed as a single growth band, or the lens flare around an eclipsed sun.
Parts of an Annulus
Before jumping into formulas, it helps to label the pieces. Every annulus is built from just three measurements:
- Outer radius (R) — the distance from the shared center to the edge of the larger, outer circle.
- Inner radius (r) — the distance from the shared center to the edge of the smaller, inner circle (the “hole”).
- Width — the thickness of the ring itself, found by subtracting the inner radius from the outer radius.
Area of an Annulus: The Formula
The logic behind the area formula is refreshingly simple: take the area of the big circle, then subtract the area of the hole.
Worked Examples
Example 1 — Finding the Area
Suppose a circular garden has an outer radius of 10 meters, with a circular fountain of radius 4 meters set into its exact center. What is the area of the planting ring (the annulus) around the fountain?
A = π(R² − r²)
= π(10² − 4²)
= π(100 − 16)
= 84π ≈ 263.9 m²
So, the gardener has roughly 264 square meters of soil to plant.
Example 2 — Finding the Perimeter
A metal washer has an outer radius of 6 cm and an inner radius of 2.5 cm. Find its total perimeter, including both edges.
P = 2π(R + r)
= 2π(6 + 2.5)
= 2π(8.5)
= 17π ≈ 53.4 cm
That 53.4 cm accounts for the full outer rim plus the full inner rim of the washer.
Example 3 — Working Backward from the Width
A circular running track forms an annulus. Its outer radius is 50 m, and the track itself is 4 m wide. What is the area of the track surface?
So, r = R − Width = 50 − 4 = 46 m
A = π(R² − r²)
= π(50² − 46²)
= π(2500 − 2116)
= 384π ≈ 1206.4 m²
Quick Reference Table
| Quantity | Formula | Example (R=10, r=4) | Result |
|---|---|---|---|
| Width | R − r | 10 − 4 | 6 units |
| Area | π(R² − r²) | π(100 − 16) | ≈ 263.9 sq. units |
| Perimeter | 2π(R + r) | 2π(14) | ≈ 87.96 units |
