Annulus Calculator
Calculate the area of an annulus from outer and inner radii. Includes circumference, width, and step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Annulus Calculator
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Formula: Area = π(R² - r²)
Worked example — Area ≈ 373.85 mm²
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.
Why does the annulus area formula subtract the two circle areas?
The area of the full outer circle is πR², and the 'hole' removed from the middle is the inner circle's area, πr². Since the annulus is everything left over once that hole is removed, its area is simply the difference: π(R² − r²), which factors as π(R+r)(R−r).
How is the annulus area formula useful in engineering, like for washers and pipes?
A metal washer's cross-section is an annulus, so π(R² − r²) directly gives the amount of material used — critical for cost estimation and weight calculations in manufacturing. Similarly, for a pipe's cross-section, the annulus formula (using the pipe's outer and inner radii) gives the wall's material area, which combined with pipe length gives total material volume.
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.
What is the area of a partial annulus, like a circular racetrack lane segment?
For a partial annulus spanning an angle θ (in radians) instead of the full 360°, multiply the full annulus area by θ/(2π): Partial Area = (θ/2)(R² − r²). This is exactly how racetrack designers calculate the area of a single curved lane section between two concentric arcs.
How is annulus area used in astronomy, like during an annular solar eclipse?
An annular solar eclipse occurs when the Moon is too far from Earth in its elliptical orbit to fully cover the Sun, leaving a bright ring (annulus) of sunlight visible around the Moon's silhouette — the term 'annular' comes directly from 'annulus.' The visible ring's apparent area in the sky can be modeled using the same π(R² − r²) formula with the Sun's and Moon's apparent angular radii.
Does the annulus formula still work if the two circles aren't perfectly concentric?
No — π(R² − r²) specifically assumes both circles share the exact same center. If the inner circle is offset (non-concentric), the resulting shape is called an 'eccentric annulus' or crescent-like region, and its area requires a more complex calculation involving circular segment overlaps rather than the simple difference of two circle areas.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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