Conic Section Calculator

What does this conic section calculator compute?

Axis-aligned standard forms: ellipse x²/a² + y²/b² = 1, parabola x² = 4py or y² = 4px, hyperbola x²/a² − y²/b² = 1 (or the vertical swap). a = b is a valid ellipse with e = 0; the circle-only page stays /calc/math/circle.

The published formula or method is x²/a² + y²/b² = 1; x² = 4py; x²/a² − y²/b² = 1. The following references support the stated mathematics or method:

• Gilbert Strang, Calculus — Chapter 3: Applications of the Derivative (MIT OpenCourseWare): Standard equations and focal geometry of axis-aligned conic sections, including translated equations and the ellipse’s semiaxes. Ellipse area and perimeter are supported by separate integral references. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
• Gilbert Strang, Calculus — Chapter 15: Vector Calculus (MIT OpenCourseWare): The planar area formula A = (1/2) ∮(x dy − y dx). For straight polygon edges it gives the shoelace expression; rectangular/square areas and trapezoid area (B+b)h/2 follow by applying it to the vertices. For x=a cos t, y=b sin t it gives ellipse area πab. Perimeters of these polygons are sums of side lengths. These formulas are derived applications of the theorem. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
• NIST DLMF — 19.30 Lengths of Plane Curves: Exact ellipse perimeter L(a,b) = 4a E(k), with k² = 1−b²/a² and a ≥ b. A displayed elementary perimeter approximation is distinct from this exact elliptic-integral definition; the reference does not establish the approximation’s error or validate CalculatorX.

Which inputs and limits should you check?

Use the indicated standard-form orientation and positive geometric parameters. A numerical circumference approximation and an exact defining equation describe different aspects of a conic.

The worked example uses a = 5, b = 4 with c = √(a² − b²). 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.conic v1.1.0 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 range, interpretation and privacy

Use one consistent length unit; area uses its square. For an ellipse, a is the x semiaxis and b is the y semiaxis. For a hyperbola, a is the transverse semiaxis, along the chosen opening direction, and b is the conjugate semiaxis. A signed nonzero p places the parabola focus and sets its opening direction. These standard forms have their center or vertex at the origin; rotated or translated conics are outside this tool.

Inputs are strict decimal numbers or decimal strings, at most 1024 characters each. Decimal text is normalized to binary64 before geometry is evaluated. Nonzero inputs and reported numeric outputs must be finite normal binary64 values, with magnitude at least 2^-1022. Nonzero text converted to zero is rejected with NUMERIC_UNDERFLOW. Nonzero subnormal inputs, underflowing outputs and overflowing outputs reject with NUMERIC_RANGE; change the length unit and retry. Empty and null fields are invalid; defaults apply only when a URL field is absent.

Circle classification uses exact equality of the normalized axes. Near a=b, the focal distance and eccentricity are sensitive to input rounding; two different decimals can round to the same binary64 value. Very thin ellipses or nearly degenerate hyperbolas can have eccentricity rounded to 1. A displayed equation is a floating point presentation, not a symbolic proof. Typical arithmetic verification uses relative tolerance 2e-12 against independent high-precision values of the stated formulas; exact zeros and discrete branches are checked separately. This tolerance does not bound sensitivity to changed inputs.

Ellipse perimeter uses Ramanujan I, π[3(a+b)−√((3a+b)(a+3b))], an approximation rather than the exact elliptic-integral arclength. Its model error grows for very thin ellipses and is separate from floating point arithmetic error. Figures use equal scale for both axes and explicitly omit aspect ratios or focal separations too small to show faithfully. Complete numerical fields remain available in JSON and CSV.

Interactive calculation runs in the browser. Shared URLs send their parameters to the server and may be server-rendered; REST/MCP requests send their inputs. A shared version identifies the requested calculation version and shows a mismatch notice when appropriate; it is not a preserved historical execution environment. Local history is stored in this browser. No zero-log, retention-period or processing-region guarantee is implied.