Circle segment area calculator

Definition

A circular segment is the region of a disk between a chord and the arc that chord subtends. For radius r > 0 and central angle θ in radians the segment area is A = (r²/2)(θ − sin θ). That is exactly the formula this page implements: area = (r·r/2)·(θ − sin(θ)), after converting degrees via θ = α·π/180 when needed.

Geometrically the segment is the sector area minus the triangular area formed by the two radii and the chord: A_sector = (r²/2)θ, A_triangle = (r²/2)sin θ, so A_segment = A_sector − A_triangle. For θ ∈ (0, 2π] the same expression covers both the minor segment (θ ≤ π) and the region for an angle larger than a semicircle.

Besides area, the result panel reports chord c = 2r sin(θ/2), sagitta s = r(1 − cos(θ/2)), arc length L = rθ, sector area, triangular area, and segment perimeter L + c.

Historical and conceptual context

Segment area sits with sector, arc, and chord in classical circle geometry. Archimedes’ Measurement of a Circle and later handbooks (CRC Standard Mathematical Tables, MathWorld — Circular Segment) give the equivalent forms A = (r²/2)(θ − sin θ) and the degree version A = (r²/2)(πα/180 − sin(πα/180)). Wikipedia (Circular segment) collects the same identities with sagitta and chord formulas.

Conceptually a segment is not a sector: the sector includes the radial triangle; the segment subtracts it. This calculator evaluates A = (r²/2)(θ − sin θ) for the entered central angle,.

In engineering the segment models channel cross-sections, CAD arc fills, and the “cap” of a circle above a chord. Arc, chord, and sagitta are companion lengths; they do not replace area A.

Formal properties

Domain: r > 0 and θ > 0 in radians after conversion (the form may accept degrees). As θ → 0⁺, A → 0. For θ = π (semicircle) sin π = 0, so A = (r²/2)π = πr²/2, exactly half the disk. For θ = 2π, sin(2π) = 0 and A = πr² — the full disk in the full-angle limit.

Identities: A = (r²/2)(θ − sin θ); A_sector = (r²/2)θ; A_triangle = (r²/2)sin θ; c = 2r sin(θ/2); s = r(1 − cos(θ/2)); L = rθ. Monotonicity: for fixed r, θ ↦ A(θ) increases on (0, 2π] because the derivative (r²/2)(1 − cos θ) ≥ 0.

Homogeneity: under r ↦ λr area scales as λ² while θ is dimensionless in radians. The degree form A = (r²/2)(πα/180 − sin(πα/180)) is equivalent via θ = πα/180.

Full derivation

Step 1. A sector of central angle θ has area proportional to the angle: A_sector / (πr²) = θ/(2π), hence A_sector = (1/2)r²θ.

Step 2. The isosceles triangle of two radii r with included angle θ has area (1/2)r² sin θ (the (1/2)ab sin γ formula with a = b = r).

Step 3. The segment belonging to that arc is sector minus triangle: A = (1/2)r²θ − (1/2)r² sin θ = (r²/2)(θ − sin θ).

Step 4. Chord: half the chord is r sin(θ/2), so c = 2r sin(θ/2). Sagitta: distance from chord midpoint to the arc is r − r cos(θ/2) = r(1 − cos(θ/2)).

Special cases

θ = π/2 (90°), r = 2: A = 2(π/2 − 1) = π − 2 ≈ 1.141593. θ = π (180°), r = 1: A = π/2 ≈ 1.570796 (semicircle). θ = π/3 (60°), r = 2: A = 2(π/3 − √3/2) = 2π/3 − √3 ≈ 0.362344.

θ = 2π/3 (120°), r = 5: A = (25/2)(2π/3 − √3/2). θ = 3π/2 (270°), r = 3: sin(3π/2) = −1, so A = (9/2)(3π/2 + 1) — the segment for an angle larger than π.

Limit θ → 0: sin θ = θ − θ³/6 + O(θ⁵) yields A ≈ r²θ³/12, matching the small-segment parabolic approximation.

Numerical and computational notes

The calculator evaluates θ in radians (sin, cos). Degrees are converted first: θ = α·π/180. Domains: r > 0; positive angle in the form’s accepted range. Float64 (floating-point arithmetic) is enough for school and engineering r; for tiny θ, cancellation in (θ − sin θ) is a classic numerical issue — on-page examples (60°–270°) stay far from that limit.

Comma and period are equivalent in the parser. Examples fill (r, angle) pairs and refresh the SVG diagram. The primary result is area A; arc, chord, and sagitta are panel metadata, not separate calculators. The diagram uses a fixed drawing radius for display only.

Related concepts and authorities

Related models: sector area A = (1/2)r²θ, arc length L = rθ, disk area πr². Segment = sector − triangle at the same θ and r. The law of cosines is not required — the angle is central, not a general SAS triangle angle.

Authorities: MathWorld — Circular Segment; Wikipedia — Circular segment; CRC Standard Mathematical Tables (Zwillinger); classical circle geometry. The calculator evaluates (r²/2)(θ − sin θ); it does not replace a textbook.