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: an arched window An arched window above a door has a 36-inch chord at the bottom and rises 18 inches to the top of the curve. That rise equals half the chord, so the arch is a full semicircle with r = 18 in.

Chord = 2r = 36 in, arc = πr = 56.55 in, total perimeter = 92.55 in of trim.

Now flatten it into a segmental arch. Keep the 36-inch chord but drop the rise to 6 inches. The radius is no longer 18: from r = (c²/4 + h²) / 2h it becomes (324 + 36) / 12 = 30 in, and the central angle falls to 2 × arcsin(18/30) = 73.7°, or 1.287 rad.

Arc = 30 × 1.287 = 38.61 in, so the perimeter is 36 + 38.61 = 74.6 in.

Same opening, 18 in less trim, and a completely different look. Note that flattening the arch made the radius bigger, not smaller, which surprises people every time: a flatter curve is a larger circle.

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.
• Bending stock for an arch. Steam-bent trim, flexible track and lead came all get cut to the arc length, not the chord, and the difference is the whole reason a piece comes up short.
• 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, for every angle. There is no exception, since the straight line between two points is the shortest path and the arc is not it.

What changes with θ is by how much. At 30° the chord is 0.518r against an arc of 0.524r, a gap of just over 1%, so on a shallow curve you can almost get away with measuring straight across. At 180° the chord is 2r and the arc is πr, and now the arc is 57% longer. The flatter the arch, the closer the two get.

Major and minor segments

A chord cuts the circle into two pieces. The one with θ < 180° is the minor segment and the one with θ > 180° is the major segment, and they share the same chord. Usually you want the minor one, so if your angle comes out above 180° you are probably measuring the wrong side and should use 360° minus your figure.

Note that this has nothing to do with which of the chord and the arc is longer. The arc wins in both cases.

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 has closed back into a full circle and the chord has shrunk to nothing. ✓