Annulus Area Calculator

Formula

A = π(R² − r²)

Worked example

With Outer radius: 8 units, Inner radius: 5 units, ring area is 122.522113 square units.

Calculation method

The rule is A = π(R² − r²).

For this annulus area example, changing outer radius from 8 units to 9 units while keeping the other inputs fixed changes ring area from 122.522113 square units to 175.929189 square units.

Reading ring area

Its area is the outer circle area minus the inner circle area, in square units. In the worked values on this page, outer radius 8 units, inner radius 5 units produce ring area 122.522113 square units.

Practical use

Use it for washers, circular paths, or ring-shaped cross-sections. The specific task here is to find ring area from outer radius, inner radius.

Limits and common mistakes

The outer radius must be at least as large as the inner radius. Enter radii, not diameters. Verify outer radius in units before interpreting ring area; a value measured on another basis would answer a different question.

Why the relationship works

An annulus is the region between two concentric circles. Its area equals the outer disk area minus the inner disk area, so both radii must be measured from the same center. Subtracting radii first and then squaring is a different expression and gives a wrong result. The annulus area rule uses outer radius (units), inner radius (units) in A = π(R² − r²); read each symbol as the field named on this page.

Interpreting another set of values

For annulus area, the fields labeled outer radius (units), inner radius (units) are the known quantities. The calculator solves for ring area in square units using A = π(R² − r²). A sector area covers a wedge, an arc length follows a curve, and an annulus area covers a ring. Use the output unit to distinguish lengths from surfaces and keep all radii centered on the same circle. When comparing two calculations, keep their units and input definitions identical.

For a second annulus area check, keep the other default inputs fixed and change outer radius from 8 units to 10 units. Ring area changes from 122.522113 square units to 235.619449 square units. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

The outer radius must be at least the inner radius, and both must use the same unit. Equal radii leave zero ring area. An annulus result has square units; its width, found by subtracting radii, is only a length and cannot replace the area formula. The reported ring area is an area. If the input lengths were centimeters, its unit would be square centimeters; the numerical 122.522113 square units therefore describes covering rather than edge length.