Quadrant of an angle calculator

This capacity accepts a finite angle in degrees or radians, reduces it with modular arithmetic into the half-open interval from zero inclusive to one full turn exclusive, classifies the terminal ray as quadrant I, II, III, or IV or names the axis, and returns structured JSON with both degree and radian normalized measures. Non-numeric or non-finite input is rejected with invalid_input and is not charged.

What quadrants mean for standard-position angles

Quadrant II continues from the positive y-axis to the negative x-axis (ninety to one hundred eighty degrees); quadrant III from the negative x-axis to the negative y-axis (one hundred eighty to two hundred seventy); and quadrant IV from the negative y-axis back to the positive x-axis (two hundred seventy to three hundred sixty, not including the endpoints).

How this calculator validates input, normalizes, and classifies

Unit defaults to deg for a full turn of three hundred sixty; set unit to rad for a full turn of two pi. The engine converts the measure to degrees for classification, computes the positive remainder modulo three hundred sixty so every real input lands in the half-open interval from zero inclusive to three hundred sixty exclusive, and snaps near-full-turn floating noise to zero for stable axis detection. When the ray is on an axis the quadrant number is omitted from the response rather than set to null; when the ray is interior the axis name is omitted. Send a JSON body with required field angle and optional field unit. Angle must be a finite number—JSON numbers and numeric strings such as one hundred thirty-five or negative forty-five are accepted; words, empty strings, null, infinity, and NaN fail with invalid_input so those failed calls are not charged. Classification then applies strict open intervals for the four quadrants and exact matches at zero, ninety, one hundred eighty, and two hundred seventy degrees for the four axes named positive_x, positive_y, negative_x, and negative_y. Results also include degrees_normalized and radians_normalized rounded to a stable twelve-decimal precision so the same input always yields the same JSON, including for irrational radian measures. Edge cases are explicit: four hundred five degrees wraps to forty-five and reports quadrant I; minus thirty wraps to three hundred thirty and reports quadrant IV; ninety degrees reports on_axis true with axis positive_y and no quadrant number.

Practical use, limits, pricing, and delivery channels

Use the browser tool when you want an immediate quadrant or axis label while checking homework, labeling a unit-circle sketch, or confirming the sign chart for sine and cosine without installing a computer-algebra system. Results are fully deterministic: the same angle and unit always yield the same classification. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted input is any finite real measure in degrees or radians; the response echoes the parsed angle and unit, reports both normalized degree and radian measures, and returns quadrant or axis fields with method quadrant_of_angle. Wire the endpoint into unit-circle demos, contest answer checkers, polar-plot preprocessing that prefers an explicit axial flag, or floating pipelines where you already prefer explicit domain errors for non-numeric payloads over opaque black-box widgets that cannot show why an angle failed to parse. It does not replace a reference-angle calculator when you need the acute measure to the x-axis, and it does not list coterminal pairs—those are separate tools with different search intent and different outputs.