Annulus area calculator

What is an annulus area calculator?

An annulus is the flat ring-shaped region bounded by two concentric circles — a larger outer circle and a smaller inner circle sharing the same center. The annulus area calculator finds the area of that ring directly from the two radii. It is essentially the area of the big circle minus the area of the hole in the middle.

This shape shows up everywhere: a washer, a pipe cross-section, a donut viewed from above, a CD, a circular running track, or the gap between two coaxial cylinders. Anytime you need to know how much surface (or how much material) lies between two circles, this calculator gives the answer in a single step.

Key concepts

• Outer radius (R) — the distance from the common center to the outer boundary of the ring.
• Inner radius (r) — the distance from the same center to the inner boundary (the hole).
• Annulus — the region between the two circles. It has two boundaries, both circular and concentric.
• Area (A) — the amount of two-dimensional surface enclosed by the annulus, measured in square length units.

How does the calculator work?

The calculator subtracts the area of the inner circle from the area of the outer circle. Because both circles share a center, the subtraction is exact — no overlap correction is needed.

Formula

$$A = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)$$

The formula requires $R > r$. If the two radii are equal, the ring collapses to a single circle of zero thickness and the area is zero. If $r > R$, the configuration is not a valid annulus, so the calculator returns no result.

You can also express the formula in terms of the ring’s thickness $w = R - r$:

$$A = \pi (R - r)(R + r) = \pi w (R + r)$$

This form is useful when you know the wall thickness of a pipe or the width of a flat ring directly.

Worked examples

Example 1: outer radius 10 cm, inner radius 5 cm

$$A = \pi (10^2 - 5^2) = \pi (100 - 25) = 75\pi \approx 235.619 \text{ cm}^2$$

Example 2: outer radius 7, inner radius 3

$$A = \pi (7^2 - 3^2) = \pi (49 - 9) = 40\pi \approx 125.664$$

The result is in the same square units as whatever length unit you used for the radii.

Example 3: equal radii

If $R = r = 5$, the ring has no width and the area is $0$. The calculator simply returns an empty result for this degenerate case.

Example 4: inner larger than outer

If you swap the values (say $R = 3, r = 7$), the configuration is not a valid annulus. The calculator returns no result rather than a negative area.

Example 5: thin ring

A washer with outer radius 12 mm and inner radius 10 mm has a thin wall of 2 mm. Using the thickness form:

$$A = \pi \cdot 2 \cdot (12 + 10) = 44\pi \approx 138.230 \text{ mm}^2$$

Practical uses

• Mechanical engineering — computing the cross-sectional area of a hollow pipe, tube, or sleeve to size flow capacity or material volume (multiply the area by length to get the volume of a hollow cylinder).
• Manufacturing — calculating material needed for washers, gaskets, flat rings, and seals stamped from a sheet.
• Architecture and landscaping — laying out circular paths, fountain rims, ring-shaped gardens, or seating around a central feature.
• Optics — measuring the clear aperture of an annular lens or stop.
• Sports — finding the area of a circular running track lane between an inner curb and outer line, complementing the circumference calculator for the lane perimeter.
• Astronomy — describing planetary rings, accretion disks, or the area of an annular eclipse’s ring of sunlight.

Notes

• Both radii must be positive, and the outer radius must be strictly greater than the inner radius.
• The result is in square units of the chosen length unit; the calculator converts automatically when you change either input or output unit.
• For a solid disc (no hole), set $r = 0$ — but in that case it is simpler to use the circle area calculator directly.
• The annulus is a 2D region. To get the volume of a hollow cylinder (an annulus extruded along an axis), multiply the annulus area by the length of the cylinder. For an elliptical version of the same idea, see the ellipse area calculator.