Sagitta calculator

A negative radius is rejected with invalid_input rather than silently taking an absolute value, so scripts and grading pipelines see an honest error when the length measure is geometrically impossible.

How to call the API and what the JSON response contains

Use it from geometry homework pipelines, arch and vault rise takeoffs, optical mirror sagitta worksheets, CNC and sheet-metal arc rise scripts, interview-prep sandboxes, or any workflow that needs reproducible sagitta numbers without installing a computer-algebra package on every machine. Send a JSON body with radius and half_angle. Field radius is the positive circle radius; aliases r and R are accepted. Field half_angle is the half central angle of the arc—the angle from the perpendicular bisector of the chord to either endpoint radius; aliases alpha, half_theta, and a map to the same meaning. Optional angle_unit is degrees by default or radians when you pass radians (aliases unit and angle_mode). On success the response echoes radius, half_angle, and angle_unit; adds half_angle_radians and half_angle_degrees for convenience; returns sagitta as the primary length result and versine as the unitless one-minus-cosine factor; and includes chord_length, distance_to_chord, central_angle_radians, central_angle_degrees, method set to sagitta_versine, and a short formula block documenting the closed forms. Classic check: radius ten and half_angle thirty degrees yields sagitta equal to ten times one minus cosine of thirty degrees, about one point three four length units, with chord length equal to ten, matching the equilateral geometry of a sixty-degree central angle. Another check: radius five and half_angle π over two radians yields sagitta five because cosine of a right half-angle is zero. Failed validation—missing a required field, a non-finite number, a negative or zero radius, an unknown angle unit, or a value outside published limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item.

Practical tips, limits, and how this differs from related tools

Half-angle may be any finite real value; cosine is even so a negative half-angle yields the same sagitta magnitude as its absolute value, which is useful when a CAD export signs the half-angle by orientation. When teaching, prefer paired examples: radius ten and thirty degrees for the equilateral-chord check against one minus square-root of three over two; radius one and sixty degrees for a short rise that students can recompute on a scientific calculator; and a negative radius to demonstrate the hard rejection path required by this product contract. Prefer this page when you already know the radius and half-angle and need the rise of the arc. Keep radius inside the published positive range up to one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Units of length are whatever you pass—metres, inches, or game units—as long as you interpret the sagitta in that same unit; the API does not convert linear units for you. Because the primary identity needs only r and α, learners can rearrange for radius when sagitta and half-angle are known, or solve for α with an inverse cosine, outside this endpoint—this tool intentionally does not invert. This landing targets people searching for a sagitta calculator or a circular-arc versine height tool; it is not a chord-length calculator (though chord length is returned as a convenience field), not a circular-segment area tool, and not an arc-length or sector-area solver—use the dedicated capabilities when those quantities are the primary search intent. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible measures while learning the contract. Round-trip checks are easy: after solving, confirm that radius minus distance_to_chord equals sagitta within floating tolerance, that sagitta equals radius times versine, and that central_angle is twice the half-angle in both units.