Arc Radius from Chord and Rise Calculator

How it works

An arc through both ends of the chord and the top of the rise belongs to exactly one circle. Its centre lies on the centreline square to the chord at its midpoint, on the side away from the arc. Half the chord, the radius, and the radius minus the rise form a right triangle, which solves to R = c² ÷ (8h) + h ÷ 2.

The trammel pivot therefore sits on that centreline, R − h from the chord midpoint, away from the arc; set the trammel to the radius and the pencil passes through both chord ends and the top of the rise. At a semicircle (rise equal to half the chord) the pivot sits on the chord itself.

The central angle is the angle the arc subtends at the pivot. The line from a chord end to the top of the rise makes a quarter of that angle with the chord, so θ = 4 × atan(2h ÷ c). The arc length, the distance along the curve, is R × θ in radians.

Formula

R            = c² ÷ (8 × h) + h ÷ 2
pivot offset = R − h = (c − 2h)(c + 2h) ÷ (8 × h)
θ            = 4 × atan(2h ÷ c)        (= 2 × asin(c ÷ 2R))
arc length   = R × θ (θ in radians)

Example

An arched rail spanning 36 in with a 3 in rise: R = 36² ÷ (8 × 3) + 3 ÷ 2 = 54 + 1.5 = 55.5 in (55-1/2 in).

The pivot sits 55.5 − 3 = 52.5 in from the chord midpoint, away from the arc. The arc subtends 37.85° and is 36.6630 in long, about 0.66 in longer than the chord.

Assumptions and limitations

• The curve is a true circular arc, symmetric about the centre of the chord; an elliptical or hand-drawn curve has no single radius.
• Chord and rise are measured to the same line, for example the pencil line or the finished edge; the radius is for that line, so a cutter or blade on the outside or inside of the line changes the cut radius by its offset.
• Rises up to half the chord (a semicircle) are covered; a larger rise makes more than half a circle, with the pivot on the arc's own side of the chord.
• Results are geometry only.

Frequently asked questions