Circle Calculator

What does this circle calculator compute?

A Euclidean circle of radius r has diameter 2r, circumference 2πr, and area πr². Arc length s = rθ (θ in radians).

The published formula or method is C = 2πr, A = πr². 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.
• Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare): Radians, the equivalence of 360° and 2π radians, and circular circumference 2πr. Scaling the unit-circle distance gives arc length s = rθ; right-triangle sine gives chord length 2r sin(θ/2). Signed angles and multi-turn behavior are CalculatorX’s explicit policy. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Keep length units consistent and distinguish circumference, curved arc length, disk area, and sector area. A degree angle must be converted to radians before using an arc or sector formula.

The worked example uses r = 1 with C = 2π, A = π. 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.circle v1.2.0 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.

Which numerical range is supported?

All nonzero outputs must be finite normal binary64 values (magnitude at least 2^-1022); exact zero angles are supported. Outside this envelope, NUMERIC_RANGE requests rescaling instead of returning zero, Infinity, or NaN. Results use 12 significant digits. Radius and arc length share the chosen length unit; areas use its square. Signed degree angles and multiple turns are preserved, while the diagram shows at most one turn.

How are request numbers and versions interpreted?

REST and MCP execute_calculation reject a nonzero raw JSON number such as 1e-400 with NUMERIC_UNDERFLOW before it becomes a zero calculation. Literal zero remains valid where the model permits it. Decimal strings preserve input text where declared, but do not extend binary64 precision; a value already sent as 0 cannot be recovered.

Every supplied calculation-version pin must agree. Conflicting header/body pins return VERSION_PIN_CONFLICT (HTTP 409); the body does not override the header. See the request contract for accepted pin locations, input-source limits, and API contract 1.0.1. Calculation-model version and request-contract version are separate.