Annulus Area Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The area of an annulus (the surface between two concentric circles) is calculated with A = π × (R² − r²), where R is the outer radius and r the inner radius. For an outer radius of 10 cm and an inner radius of 6 cm, the area is about 201.06 cm².
Explanation
An annulus (also called a circular ring) is the plane surface left when a disk of radius r is removed from the center of a larger disk of radius R — the typical shape of a washer, a CD, or a roundabout seen from above. Its area is calculated simply as the difference between the area of the large disk and that of the small removed disk: since the area of a disk of radius r is π × r² (see our circle area and perimeter calculator), the area of the annulus becomes π × (R² − r²) directly, not π × (R − r)² as one might be tempted to write by mistake — these two expressions are equal only in the special case where r = 0. The formula also gives consistent results at both limiting cases: when r tends to 0, the annulus becomes a full disk and the area tends to π × R²; when r approaches R, the annulus thins out and its area tends to 0, as the test case where R = r confirms. This same area-subtraction logic appears in other compound figures, for example to calculate the actually painted area of a wall with a circular window, or to estimate the amount of material needed to make an annular mechanical part — an engineering problem where the regular polygon area calculator can also come in for similar non-circular shapes.
Example: outer radius 10 cm, inner radius 6 cm
Inputs
Outer radius: 10 cm. Inner radius: 6 cm.
Calculation
A = π × (10² − 6²) = π × (100 − 36) = π × 64 ≈ 201.06 cm².
Result
This annulus has an area of about 201.06 cm².
Frequently asked questions
Why not just calculate π × (R − r)²?
Because that expression does not correspond to the area of the annulus: it would calculate the area of a disk whose radius is the difference of the two radii, a geometrically different figure. The right approach is to subtract the two disk areas separately (πR² − πr²), which factors into π(R² − r²) — an expression different from π(R−r)² as soon as r is not zero, the gap between the two growing with the size of the inner radius.
What happens if the inner radius is larger than the outer radius?
The formula is still computable but then gives a negative result with no physical meaning — this case corresponds to inconsistent input (the small circle cannot exceed the large circle) rather than a true mathematical limit of the formula itself. The inner radius must always stay less than or equal to the outer radius for the result to represent a real area.
Does this formula apply to an annular sector (a portion, not the whole ring)?
Not directly: this calculator gives the area of the whole annulus, over the full 360° of the circle. For a portion of an annulus bounded by an angle, you would multiply this result by the fraction that angle represents out of 360°, on the same principle as in our circular sector area calculator for a full disk.