Circle sector area calculator

Definition

A circular sector is the portion of a disk bounded by two radii and the arc between their endpoints. For radius r > 0 and central angle θ in radians the sector area is A = (1/2)r²θ. Equivalently in degrees α: A = πr²·(α/360). This calculator computes A = 0.5·r·r·θRad after degree conversion, matching calculator-pole-wycinka-kola.js.

The sector is the “wedge” of the circle: at θ = 2π (α = 360°) it recovers the full disk πr²; at θ = π it is a semicircle. Unlike a segment, the sector includes the radial triangle. The panel also shows arc L = rθ, chord, disk fraction θ/(2π), and segment area A − (1/2)r² sin θ as metadata.

The area unit follows r. Accepted angle range on this page: 0 < θ ≤ 2π (0° < α ≤ 360°).

Historical and conceptual context

Sector area follows from proportionality of disk area to central angle — an ancient fact recorded in handbooks (MathWorld — Circular Sector; Wikipedia — Circular sector; CRC Standard Mathematical Tables). The radian form A = (1/2)r²θ is canonical in analysis because radian measure makes the coefficient simple.

Conceptually the sector is the set of disk points whose polar angle lies in an interval of length θ. In CAD and statics it models wedge loads, sections, and chart sectors. This page does not take segment area as the primary result — that lives on the segment calculator.

Equivalence A = (1/2)r²θ = πr²·(α/360) follows from θ = α·π/180: (1/2)r²·(απ/180) = πr²·α/360.

Formal properties

Domain: r > 0, θ ∈ (0, 2π]. Codomain: A ∈ (0, πr²]. For fixed r the map θ ↦ A is linear: A = (r²/2)θ. Homogeneity in r: A(λr, θ) = λ² A(r, θ).

Identities: A = (1/2)r²θ; A = πr²·(α/360); L = rθ; fraction f = θ/(2π) = α/360; A_segment = A − (1/2)r² sin θ; chord c = 2r sin(θ/2) (at θ = 2π the page sets c = 0).

For θ = π/2, A = πr²/4 (quarter disk). For θ = π, A = πr²/2. Strict monotonicity in θ holds on (0, 2π].

A sector of angle θ is also the polar integral ∫₀^θ ∫₀^r ρ dρ dφ = (1/2)r²θ. That analytic derivation confirms the school proportion to disk area and matches A = 0.5·r²·θRad.

The check 0 < θ ≤ 2π blocks multi-turn angles that would algebraically yield A > πr² for θ > 2π; this page does not compute that excess. A full turn θ = 2π is allowed and returns A = πr².

Degree and radian modes are linked by θ = α·π/180 before the half-r-squared product. Comma and period parse identically.

Full derivation

Step 1. The full disk has area πr².

Step 2. A full turn is 2π rad = 360°. A sector of angle θ occupies the fraction θ/(2π), so A = πr²·θ/(2π) = (1/2)r²θ.

Step 3. In degrees α the fraction is α/360, hence A = πr²·(α/360). Substituting θ = απ/180 links both forms.

Step 4. Arc length: circumference 2πr times θ/(2π) gives L = rθ — the same proportion.

Worked checks: α = 90° with r = 2 yields A = π; α = 180° with r = 1 yields A = π/2; α = 360° recovers the full disk. Those three anchors verify both the radian product (1/2)r²θ and the degree form πr²·(α/360) before any diagram is trusted.

Special cases

α = 90°, r = 2: θ = π/2, A = π ≈ 3.141593. α = 180°, r = 1: A = π/2. α = 360°, r = 1: A = π. α = 60°, r = 5: A = 25π/6 ≈ 13.089969.

α = 120°, r = 3: A = 3π ≈ 9.424778. α = 45°, r = 4: A = 2π ≈ 6.283185. Radian input θ = 1.570796 with r = 2 recovers the quarter disk.

As θ → 0⁺, A → 0 linearly (unlike the segment, which vanishes like θ³).

Numerical and computational notes

The calculator takes θ = α for radians or θ = α·π/180 for degrees, then computes A = 0.5·r²·θ. Validation rejects r ≤ 0 and angles outside (0, 2π]. Float64 is enough; products with π inherit π’s approximation.

Examples cover quarter, half, and full disks. The SVG uses a fixed drawing radius; visual scale does not change A. Comma and period are equivalent.

Together with arc length L = rθ one obtains A = (1/2)r L — the “curvilinear triangle” with base L and height r. For small θ the arc is nearly a chord and A still scales linearly in θ, whereas segment area vanishes faster.

Panel metadata (chord, segment, disk fraction) share the same θ and r; the primary result remains sector area. The SVG uses a fixed drawing radius only to visualize the wedge.

Quarter-disk (90°, r = 2 → A = π), semicircle, and full-turn examples confirm A ∝ θ at fixed r. Segment area on the panel is informational only; the dedicated segment calculator is the companion page.

Related concepts and authorities

Related: segment area A = (r²/2)(θ − sin θ), arc length L = rθ, disk area πr². Sector minus triangle = segment.

Authorities: MathWorld — Circular Sector; Wikipedia — Circular sector; CRC / Zwillinger; classical circle geometry. The calculator reports A = (1/2)r²θ.