Annulus Calculator

What does this annulus calculator compute?

An annulus of outer radius R>r and inner radius r>0 has area π(R²−r²) and perimeter 2π(R+r). R=r is a circle, not an annulus.

The published formula or method is A = π(R² − r²), P = 2π(R + r). The following references support the stated mathematics or method:

• Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare): Polar coordinates and the circular wedge area r²Δθ/2, together with area = ∫r² dθ/2 and polar arc-length integration. A full turn gives disk area πr² and circumference 2πr. An annulus follows by subtracting disk areas and adding both boundary lengths; these are derived applications. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Enter positive radii with R > r. The perimeter includes both boundary circles. Area uses squared length units, while perimeter uses the radius’s length unit.

The worked example uses R = 2, r = 1 with A = 3π. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.plane v1.6.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical domain, polygon order and reproducibility

Use one consistent length unit; area uses its square. Side/base lengths, radii and heights must be positive normal binary64 numbers. Angles are degrees strictly between0 and180; degree-to-radian conversion must remain normal. Regular n is a safe integer3 through9007199254740991. The annulus requires R>r>0; its perimeter includes both circles. The trapezoid model is isosceles, including the equal-base rectangle case.

For irregular polygons provide3–8 ordered distinct vertices as x,y; x,y, a flat even coordinate list, [x,y] arrays or {x,y} objects in machine calls. An optional repeated first vertex closes the ring. Clockwise and concave polygons are supported. Exact predicates on normalized binary64 reject self-intersections, nonadjacent touching/overlap, duplicate vertices, adjacent backtracking and zero area; straight intermediate vertices without overlap are allowed.

Strict decimal text is limited to1024 characters per scalar (32768 for a vertex list); unknown fields and conflicting active aliases are rejected. Nonzero text converted to zero returns NUMERIC_UNDERFLOW. Nonzero inputs and all positive output measures must be finite normal binary64 (at least2^-1022); otherwise NUMERIC_RANGE requires changing units or translating coordinates. Exact zero coordinates are supported.

Polynomial measures use exact dyadic intermediates, scaled norms and stable half-angle diagonals. Annulus area uses the factored difference; polygon area and topology use exact binary64 predicates. Transcendental functions remain binary64. No absolute small-value clamp or display rounding replaces a nonzero result with zero. Thin rings, nearly flat shapes and tiny offsets relative to large coordinates are sensitive to input rounding: predicates concern normalized values, not the original arbitrary-precision decimal geometry. Relative1e-11 verification compares arithmetic with independent high precision; it does not bound input uncertainty.

Figures use equal coordinate scales. A thin, nearly collapsed or unresolvable figure, or a regular polygon with more than128 sides, is explicitly omitted while complete numerical results remain readable and exportable. This bound keeps rendering independent of a very large n.

Interactive calculation runs locally. A shared URL sends parameters to the server for replay/SSR and may enter browser history, access logs or analytics. REST/MCP send inputs for server execution. Site analytics may receive page URLs and interaction metadata. No zero-log, fixed retention, deletion or processing-region guarantee is claimed.