How to use this calculator
- Span the curve. Lay a straightedge or rule across the arc — its length between contact points is the chord.
- Measure the rise. The gap from the straightedge to the arc at mid-span, square to the chord. Use an indicator on shallow curves.
- Compute. r = c²/8h + h/2 gives radius and diameter; angle and arc length follow.
- Cross-check. Repeat with a different chord length — agreement between two spans confirms the surface really is circular.
How it works
A chord and its rise fully determine a circle — the intersecting-chords theorem applied to the diameter through the midpoint:
r = c²/(8h) + h/2 · h = r − √(r² − c²/4) · θ = 2·arcsin(c/2r) · s = rθ
The relation is exact for any rise, including major arcs (rise larger than the radius). It is the geometry behind radius gauges, layout templates and roll-bending checks — and the inverse problem of the ball-nose stepover calculator, where the same sagitta is the scallop left between passes. Bend-allowance lengths for formed parts live in the bend allowance calculator, and pipe versions of the same layout in the pipe bend developed length calculator.
Worked example
Verified against the live calculator
A template curve spans a 24 in chord with a
3 in rise at mid-span:
r = 24²/(8 × 3) + 3/2 = 24 + 1.5 = 25.5 in → Ø51 in
The included angle is 2·arcsin(12/25.5) ≈ 56.1°, and the material along the curve measures 25.5 × 0.980 ≈ 25.0 in — an inch longer than the chord, which is exactly the stretch a flat blank has to supply when the curve gets rolled or the template gets banded.
Frequently asked questions
How do you find the radius of an arc from a chord and height?
r = c²/(8h) + h/2 — exactly, not approximately. Lay a straightedge across the curve (the chord), measure the gap at its middle (the rise or sagitta), and a 24 in chord with a 3 in rise gives r = 576/24 + 1.5 = 25.5 in.
What is the sagitta of an arc?
The height of the segment: the perpendicular distance from the midpoint of the chord to the arc — h = r − √(r² − c²/4) for a minor arc. It is the gap a feeler or indicator reads under a straightedge laid on a curved surface.
How do you find the diameter of a broken wheel or circle fragment?
Three points define a circle, and chord-plus-rise is the bench version of three points. Span the fragment with a rule, measure the bulge at the middle, and r = c²/8h + h/2 recovers the original radius — the classic trick for broken flywheels, gaskets and gear blanks.
Why is the radius so sensitive on shallow arcs?
Because h appears in the denominator: for a 40 in chord with a 1/4 in rise, r = 1600/2 + 0.125 ≈ 800 in, and being off by 0.025 in on the rise moves the radius by about 80 inches. Long chords need indicator-grade rise measurements, not a rule.
Method & assumptions
- Exact circular-segment geometry (intersecting-chords theorem); no small-angle approximation anywhere.
- Rise larger than the radius is handled as a major arc — angle and arc length report the long way around.
- The shallow-arc warning fires below a 1% rise-to-chord ratio, where measurement error in h dominates the answer.
- Assumes the surface is actually circular — two different chord spans agreeing is the practical test; ovality splits them.