Hollow Circle Area Calculator

In Euclidean geometry, a hollow circle is called an annulus (plural: annuli, from the Latin word for 'little ring'). It is the planar 2D region enclosed between two concentric circles sharing the same central origin but having different radii.

Using radii: Area = π × (R² - r²), where R is the outer radius and r is the inner radius. Using diameters: Area = (π / 4) × (D² - d²) ≈ 0.785398 × (D² - d²), where D is the outer diameter and d is the inner hole diameter.

Because (R - r)² = R² - 2Rr + r², which is NOT equal to (R² - r²). Subtracting first gives you the area of a circle with radius equal to the wall thickness, which vastly underestimates the true ring area. You must square each radius individually before subtracting.

Since (R² - r²) = (R - r)(R + r), we can write Area = 2π × r_avg × t, where t = (R - r) is the radial wall thickness and r_avg = (R + r) / 2 is the mean centerline radius. This is equivalent to unrolling the ring into a rectangle of length 2π r_avg and width t.

Discovered by mathematician Mamikon Mnatsakanian, this theorem states that the area of any annulus equals the area of a circle whose diameter is equal to a tangent chord (L) touching the inner circle and bounded by the outer circle: Area = (π × L²) / 4. Remarkably, you don't even need to know R or r!

A hollow circle has two distinct physical boundaries: the outer perimeter (2πR) and the inner perimeter (2πr). Its total boundary length is the sum of both: P_total = 2π(R + r) = π(D + d).

Multiply the hollow cross-sectional area by the height or axial length of the cylinder: Volume = π × (R² - r²) × Length = (π / 4) × (D² - d²) × Length.

For torsional mechanical stress, the polar moment of inertia is J = (π / 2) × (R⁴ - r⁴) = (π / 32) × (D⁴ - d⁴). Hollow shafts provide high torsional rigidity with substantial weight savings compared to solid shafts.

Divide the total bolt clamping load force (F) by the annular gasket contact area (A): Sealing Stress σ = F / [π(R² - r²)]. Engineers compare this stress against minimum seating yield values (y) to prevent high-pressure pipeline leaks.

An annulus is a flat two-dimensional surface (like a flat steel washer or a compact disc). A torus is a three-dimensional doughnut-shaped volume formed by revolving a small circle around an external axis.

First calculate the cross-sectional area A = π(R² - r²). Multiply by the axial thickness (h) to get volume (V = A × h). Finally, multiply volume by material density (e.g., steel density ≈ 7.85 g/cm³ or 490 lb/ft³).

You can input outer and inner dimensions in millimeters (mm), centimeters (cm), meters (m), inches (in), or feet (ft). The resulting area is provided in the matching square units.