Reproducing the answer
Square both radii, subtract inner from outer, and multiply by π.
Math calculator
Annulus Area Calculator
Calculate the area between two concentric circles. Each submitted value produces annulus area plus the intermediate reasoning.
Annulus Area inputs
Start with the given values
Seeing the method in action
Outer radius 10 and inner radius 6 give area 64π≈201.062.
Problems this can answer
It measures rings, washers, pipe cross-sections, circular borders, and material removed by a centered hole. A related application of Annulus Area is other plane areas.
The idea behind the result
Annulus area is π(R²−r²), the outer disk minus the inner disk. Annulus Area also connects with ring boundaries.
The calculation treats outer radius and inner radius according to their labels, not as interchangeable values in annulus area.
Before relying on the answer
Both radii must be nonnegative and the outer radius cannot be smaller than the inner radius.
An impossible annulus area sign or magnitude usually points to field assignment before it points to rounding.
Using the result beyond this page
Start the annulus area setup by pairing every source number with outer radius and inner radius. Convert unlike units before typing, and postpone rounding until the displayed annulus area is ready to report.
Increasing outer radius has a quadratic effect; increasing inner radius removes area quadratically. Use that direction of change to check the displayed annulus area before copying it elsewhere.
When copying the result elsewhere, include its label and any squared, linear, angular, or percentage unit implied by the inputs. That record distinguishes a calculated annulus area from an unlabeled number and makes later checking substantially easier.
A circle is the special case with inner radius zero. Keep that boundary in mind when interpreting the numerical result.
For later verification, record outer radius and inner radius before rounding the annulus area. The unrounded working value can feed subsequent steps while the rounded value serves presentation.
How to audit the Annulus Area result
Factor R²−r² as (R−r)(R+r) for a second calculation. This form separates ring thickness from its average diameter scale. If the inner radius approaches zero, the expression should smoothly return the area of the outer circle.
For Annulus Area, a useful audit records the formula, the supplied quantities, and one relationship that the answer must satisfy. Carry extra digits through that relationship and round only the final comparison. This separates numerical display differences from a genuine setup error.
Questions about Annulus Area
Can the inner radius be zero?
Yes, producing a disk.
Why must the circles be concentric?
The simple radius-difference formula assumes a shared center.
What units are used?
Square units.