Annulus Area CalculatorFree online math calculator — instant results, no sign-up
Calculate the area between two concentric circles.
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About the Annulus Area Calculator
The Annulus Area Calculator finds the ring-shaped area between two concentric circles by subtracting the inner circle’s area from the outer circle’s area. Enter the outer and inner radii in the same unit to obtain the annulus area in square units. With the default outer radius of 5 and inner radius of 3, the result is 50.2654824574 square units.
- Outer radius
- The distance from the shared center to the outer circle’s boundary, defaulting to 5, measured in a length unit such as centimeters or meters.
- Inner radius
- The distance from the shared center to the inner circle’s boundary, defaulting to 3, measured in the same length unit as the outer radius.
Formula and method
A = π(R² − r²); with the default inputs, A = π(5² − 3²) = π(25 − 9) = 16π ≈ 50.2654824574.
- AArea of the annulus, expressed in square units.
- ROuter radius, expressed in the chosen length unit.
- rInner radius, expressed in the same length unit as R.
- πPi, the ratio of a circle’s circumference to its diameter.
Square each radius, subtract the inner radius squared from the outer radius squared, and multiply the difference by π. For the defaults, squaring 5 and 3 gives 25 and 9; their difference is 16, so the exact area is 16π and the displayed decimal result is 50.2654824574.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Outer radius5
- Inner radius3
Results
- Result50.2654824574
How to use the Annulus Area Calculator
- 1Choose one length unit for both radii and convert any measurements that use a different unit.
- 2Enter the outer radius in Outer radius; the default is 5.
- 3Enter the inner radius in Inner radius; the default is 3.
- 4Check that both radii are nonnegative and that the inner radius does not exceed the outer radius, then run the calculation.
- 5Read the Result as an area in square units; the default inputs give 50.2654824574.
Reading your result
The Result row reports the area inside the outer circle but outside the inner circle; it does not include the central hole. With outer radius 5 and inner radius 3, Result = 50.2654824574 square units. If the radii are measured in centimeters, the result is in cm²; if they are in meters, it is in m².
Limitations
The calculation assumes two perfect circles in a flat plane with the same center and radii measured in the same unit. Valid radii satisfy 0 ≤ r ≤ R; a genuine ring with a central hole requires 0 < r < R. Negative radii or an inner radius larger than the outer radius are not valid annulus dimensions. The decimal result is rounded, and its displayed precision does not imply that the input measurements are equally precise.
Common questions
Can I enter diameters instead of radii?+
Not directly. Divide each diameter by two before entering it in the corresponding radius field; entering diameters as radii would make the calculated area four times too large.
Why is the formula not π(R − r)²?+
R − r is the ring’s thickness, not the radius of a circle with the same area. The annulus area is the difference between two circle areas: πR² − πr². With the default radii 5 and 3, that difference is 16π ≈ 50.2654824574.
What happens if the inner radius is zero or equals the outer radius?+
If the inner radius is zero, there is no central hole and the formula gives the full outer circle’s area. If the radii are equal, the ring has no thickness and its area is zero.
Does this calculator give the ring’s perimeter?+
No. It gives area only. For an annulus with a central hole, the total boundary length is the sum of the outer and inner circumferences, 2πR + 2πr.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.