An annulus — the ring-shaped region between two circles sharing a center — is fully defined by just its outer and inner radii. From those two numbers you can compute the ring's area, its total perimeter, and its width, or work backward to find the radius needed to hit a target area. This calculator handles all of it instantly.
How the Annulus Calculator works
The area formula follows directly from subtracting one circle's area from another: A = πR² − πr² = π(R² − r²). Perimeter adds the circumferences of both boundary circles together: P = 2πR + 2πr = 2π(R + r). Width is simply the radial gap between the two circles, w = R − r. All three formulas are exact — there is no approximation involved, unlike some other curved shapes.
Inputs and what they mean
Enter the outer radius (R), the distance from the center to the ring's outer edge, and the inner radius (r), the distance from the center to the edge of the hole. Both must be positive, and the inner radius must be strictly less than the outer radius — otherwise there is no ring to measure. On the Solve tab, provide a target area and a fixed outer radius to back-calculate the inner radius that produces it.
Limits and edge cases
If the inner radius equals or exceeds the outer radius, the shape is invalid (either zero area or a negative one) — the calculator flags this rather than returning a nonsensical number. When solving for the inner radius, if the target area is greater than or equal to the full outer circle's area (πR²), no valid inner radius exists, since the ring can never cover more area than the whole circle it's cut from. This calculator models a flat 2D annulus; for a 3D ring-shaped solid (like a pipe's cross-section extruded into a tube), you would also need a length or height dimension.