Annulus Calculator
Calculate the area of an annulus from outer and inner radii. Includes circumference, width, and step-by-step formulas.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Area = π(R² - r²)
An annulus is the ring-shaped region between two concentric circles of outer radius R and inner radius r. Its area is the outer circle's area minus the inner circle's area: π(R² − r²), which factors as π(R+r)(R−r).
Worked Examples
Example 1: Metal washer material
Problem:A steel washer has an outer radius of 12 mm and an inner (bolt hole) radius of 5 mm. Find its area.
Solution:Area = π(R² − r²) = π(12² − 5²) = π(144 − 25) = 119π ≈ 373.85 mm².
Result:Area ≈ 373.85 mm²
Example 2: Circular garden path
Problem:A circular flower bed has a radius of 4 meters, surrounded by a walking path out to a radius of 5.5 meters. Find the area of the path (the annulus).
Solution:Area = π(R² − r²) = π(5.5² − 4²) = π(30.25 − 16) = 14.25π ≈ 44.77 m².
Result:Path area ≈ 44.77 m²
Frequently Asked Questions
What exactly is an annulus?
An annulus is the flat, ring-shaped region between two concentric circles (circles sharing the same center) — think of a washer, a CD, a donut viewed from above, or a circular racetrack lane. It is defined entirely by two measurements: the outer radius R and the inner radius r, where r must be smaller than R.
Can the annulus width alone (not radii) be used to find the area?
If you know the constant width w = R − r and the mean radius r_m = (R + r)/2, the area simplifies to Area = 2π × r_m × w — essentially treating the annulus as a thin rectangular strip wrapped into a ring. This form is common in mechanical engineering when tolerances are specified as a wall thickness rather than as two separate radii.
How does an annulus differ from a circular sector or a circular segment?
An annulus is bounded by two full concentric circles (a complete ring). A circular sector is a 'pie slice' bounded by two radii and an arc of one circle. A circular segment is the region cut off by a chord and an arc. All three are common circle-based area problems, but only the annulus involves two different radii from the same center.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy