What Is an Arc Length Calculator?

An arc length calculator computes the arc length, chord length, sector area, segment area, and apothem of a circular sector. Pre-filled with radius=5 and angle=72° (1/5 of a circle): arc length = 2π ≈ 6.283, sector area = 5π ≈ 15.708 sq units. The formula is s = rθ where θ must be in radians.

Arc Length Formula

s = r × θ (θ in radians). Since 360° = 2π radians, arc length = r × 2π × (degrees/360) = πd × (degrees/360) where d = diameter. For a semicircle: s = πr. For a quarter circle: s = πr/2. The proportional relationship makes sense: doubling the radius doubles the arc length; doubling the angle doubles the arc length.

Sector vs Segment Area

Sector area = (1/2)r²θ — the pie-slice shape including the two radii and arc. Segment area = (1/2)r²(θ − sin θ) — the crescent between the chord and arc (sector minus the triangle). The triangle area = (1/2)r² sin θ. For small angles, sin θ ≈ θ, so segment area ≈ 0. For θ = π (semicircle): segment = (1/2)r²(π − 0) = πr²/2 = half the circle area.

Chord Length Formula

Chord = 2r × sin(θ/2). A chord is the straight-line distance across the arc. For θ = 180° (semicircle), chord = 2r × sin(90°) = 2r = diameter ✓. For θ = 60°: chord = 2r × sin(30°) = 2r × 0.5 = r — an equilateral triangle inscribed in the circle has sides equal to the radius. For small angles, chord ≈ arc (both ≈ rθ).

Applications in Road Design

Horizontal curves in road design are circular arcs. Engineers specify curves by their radius (R) and degree of curvature (D° — the central angle subtended by a 100-foot arc). Arc length between stations tells how far the road curves. Superelevation (banking angle) is calculated from the design speed and radius. The sight distance around a curve is the arc length visible to the driver.