Circular arc

A connected portion of a circle's circumference between two endpoints, specified by its center, radius, orientation, and central-angle span.

Version

v1 · 2026-09-28 · History

Domain-specific #

8461

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Euclidean Geometry → Mathematics

Core Idea

A circular arc is one connected path along a circle's circumference between two endpoints. Center, radius, path orientation, and central angle determine it; its length is rθ when θ is in radians. Distinguishing minor from major arcs prevents the same endpoint pair from being mistaken for a unique path, while a semicircle is the equal-path limiting case. The center and endpoints define a central angle θ. The center and endpoints define a central angle θ.

How would you explain it like I'm…

A Piece of Circle Edge

Draw a circle, then pick two spots on its edge. The curvy bit of the edge between those two spots is a circular arc. There are two ways around, a short way and a long way, so you have to say which one you mean.

Piece of a Circle's Edge

A circular arc is part of the curved edge of a circle, between two points on it. Most of the time there are two ways to go between those points: the short way (the minor arc) and the long way (the major arc), so you have to say which one you mean. The bigger the angle the arc makes at the center, the longer the arc. An arc isn't the straight line joining its ends (that's called a chord), and it isn't the pizza-slice shape (that's a sector).

Arc of a Circle

A circular arc is a connected piece of a circle's circumference between two distinct endpoints. Unless the endpoints are exactly opposite each other, they split the circle into a shorter minor arc and a longer major arc, so you need to say which one you mean, for example by giving an angle and a direction. The angle made at the center by the two endpoints is the central angle θ. If θ is in radians, the arc length is L = rθ; in degrees α, L = απr/180. Either way, the arc gets the same fraction of the full circumference as its angle is of a full turn. An arc is different from its chord (the straight segment between the endpoints) and from a sector (the region bounded by two radii and the arc).

 

A circular arc is a connected subset of a circle's circumference between two distinct endpoints. For non-antipodal endpoints there are two candidates, the minor arc and the major arc, and a choice must be made, either by naming which or by giving an oriented central angle. The center and endpoints determine a central angle theta; with theta in radians the arc length is L = r theta, and with degree measure alpha it is L = alpha pi r / 180. Both formulas express proportionality: the arc takes the same fraction of the full circumference 2 pi r as its angle takes of a full turn. The arc should be distinguished from its chord, the straight segment joining its endpoints, and from the sector bounded by the two radii and the arc. Edge cases need explicit conventions: when the endpoints are diametrically opposite the two arcs are semicircles, and zero-span and full-circle cases are ambiguous if described only by endpoints.

Scope of Application

Circular arcs occur wherever geometry or design uses constant-radius boundary segments with declared endpoints and orientation. Use it for exact constant-radius geometry only after specifying supporting circle, endpoints, minor/major or orientation choice, angle unit, and degenerate-case convention. Keep the curved path distinct from its straight chord, the bounded sector, and a noncircular segment with the same endpoints.

• Euclidean geometry. Relates central angle, chord, sector, and circumference.
• Technical drawing. Specifies constant-radius profiles.
• Road and rail alignment. Models horizontal circular curves under engineering constraints.
• Computer graphics. Represents arc primitives and paths.
• Manufacturing. Dimensions curved edges and toolpaths.

Clarity

The abstraction forces four pieces often omitted in casual language: supporting circle, endpoints, path choice, and angle unit. It prevents chord length or degree measure from being inserted into L=rθ without conversion. The closest near miss sets the boundary: The chord is the closest near miss because it shares endpoints but follows the straight interior segment rather than the circumference.

Manages Complexity

Infinitely many points on a curved path collapse to center, radius, endpoints, and orientation. Those parameters make exact length and construction possible while preserving the degeneracy cases that a simple 'curved line' description loses. The central endpoint simplicity–path ambiguity tradeoff is this: Two endpoints usually select two circumference paths. A second degree familiarity–radian naturalness tension matters because Degrees are intuitive while the length formula is direct only in radians.

Abstract Reasoning

Use three linked moves: identify the supporting circle and radius; fix two endpoints and choose minor, major, or oriented traversal; measure the corresponding central angle in radians. As a collapse test, the case exits when curvature is not constant about one center, the selected path is disconnected, or endpoint/orientation conventions do not determine a unique portion. A fourth check is to compute length by L=rθ and compare with chord only when relevant. A final check is to state conventions for diametric, coincident, or full-circle endpoints.

Knowledge Transfer

The definition transfers literally across pure geometry, CAD, mapping, and engineering when a true constant-radius segment is used. Calling any narrative or career path an arc is metaphor. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. A circular arc is a constant-curvature curve segment. Arc length is derived from radius and angle.

Relationships to Other Abstractions

Current abstraction Circular arc Domain-specific

Foundational — no parent edges in the catalog.

Children (1) — more specific cases that build on this

• Sagitta Domain-specific presupposes Circular arc

  A sagitta is defined relative to a selected circular arc and its chord.

Neighborhood in Abstraction Space

Circular arc sits in a sparse region of the domain-specific corpus (62^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Domain-Specific Measurement Parameters (36 abstractions)

Nearest neighbors

• Central angle — 0.86
• Supplementary Angles — 0.85
• Rhumb line — 0.85
• Ribbon Theory — 0.84
• ARGUS distribution — 0.84

Computed from structural-signature embeddings · 2026-10-08