Circular Segment Perimeter Calculator

The perimeter of a circular segment is the chord (the straight line) plus the arc (the curved line) that together bound the segment.

P = 2r × sin(θ/2) + r × θ

θ must be in radians. The first term is the chord length; the second is the arc length.

Worked example — arched window: An arched window above a door has a 36-inch-wide chord at the bottom and an 18-inch arc rise (the sagitta) at the top. With r = 18 in for a half-moon-style top.

If the top is a full semicircle (θ = π = 180°): chord = 2r = 36 in, arc = π × r ≈ 56.55 in. Total perimeter = 92.55 in of trim.

For a flatter “segmental arch” instead — say the same 36-inch chord but only a 6-inch rise — the central angle is smaller (about 78°). Trim length drops to about 36 + 38.4 ≈ 74 in. Same chord, much less curved trim, very different look.

Where segment perimeter matters in practice:

• Arched-window trim. Painters and finish carpenters estimate molding lengths around the curved tops of doors and windows. Add 10% to your calculation for mitre cuts.
• Tunnel lining or pipe interior. Drainage culverts and partially-filled pipes need lining material along the wet perimeter.
• Saw cuts for arched lumber. A jigsaw cutting an arch from plywood is following the arc — knowing arc length helps estimate blade life.
• Garden bed edging at corners. A planting bed cut into a curve against a fence uses segment perimeter for the edging material.

Chord vs. arc — the gotcha:

The chord is always shorter than the arc. The ratio depends on θ. For small angles, they’re nearly equal — at 30°, chord ≈ 0.518 × r and arc ≈ 0.524 × r, a difference under 2%. At 180°, chord is 2r but arc is π × r ≈ 3.14r — a 57% difference. The flatter the arch, the closer chord and arc lengths get.

Connecting back to the segment area:

A segment that has more arc than chord (θ > π) is technically a “major segment” — usually you mean the smaller piece. If you find yourself with θ > π, you might be measuring the wrong side of the chord.

Sanity check:

• θ → 0: P → 0 (the segment vanishes). ✓
• θ = π: chord = 2r, arc = πr, perimeter = 2r + πr (matches the semicircle). ✓
• θ = 2π: chord = 0, arc = 2πr (the segment closes back to a full circle, no chord — degenerate). ✓