Circular Segment Perimeter Calculator
Compute the perimeter of a circular segment — chord length plus arc length.
For arched openings, tunnel sections, and partial fills.
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). ✓
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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