Three arc equations, one baseline
Chord span W joins the endpoints; rise H is the crown’s height above the chord; radius R is measured from the circle center. For a minor arc, R = W²/(8H) + H/2, H = R − √(R² − (W/2)²), and W = 2√(2RH − H²). The calculator uses a numerically stable form of the rise equation for shallow arcs. The circle center lies R − H below the chord midpoint. Central angle is 2 asin(W/(2R)) and curved length is R × angle in radians.
For a 36 in span and 6 in rise, the radius is 30 in. Center a 30 in trammel/pivot 24 in below the chord’s midpoint, or use ordinate stations. The arc length is about 38.6 in; that is not the 36 in chord length. Do not use this for a pointed, elliptical, catenary, or greater-than-semicircle arch.
From stations to a workpiece
At each selected X from the left chord end, the rise is Y = H − R + √(R² − (X − W/2)²). The implementation rationalizes the square-root difference so very shallow curves retain useful precision. Mark the chord straight, verify its midpoint, mark rise perpendicular to that baseline, then fair a flexible batten through the points or trace the print line. Equal-arc stations space positions along the circular curve rather than the chord.
Values are theoretical centerline geometry; kerf, tool diameter, stock allowance, edge profiles, wood movement, and springback are not included. For critical work, dry-fit or test the curve.
Printing at physical size
Each template page uses physical millimeter CSS dimensions inside an A4/Letter CSS page box with 10 mm page margins. Adjacent drawing windows overlap by 10 mm, with matching registration crosses. The 100 mm ruler line checks the actual print, not just the file. Adobe’s PDF printing guidance distinguishes “Actual size” from “Fit,” which can change physical dimensions. Even a correctly generated PDF can be rescaled by a printer driver or viewer. Use matching paper, disable scaling, and measure before cutting.
Template printing is capped at 32 sheets so an accidental unit mistake does not produce hundreds of pages. Larger arcs still provide stations and CSV; subdivide the project or mark them with a trammel.
Frequently asked questions
Is span the same as arc length?
No. Span is the straight chord between endpoints. Arc length follows the circular line and is always greater unless the rise approaches zero.
Why was my rise rejected?
For a minor circular arc, the crown is at most half the span above the chord. A higher rise implies a major/horseshoe arc, which is not a single Y value for every chord X and needs a different template.
How do I know the print is really 1:1?
Print on the selected A4 or Letter sheet at Actual size / 100%, without Fit. Measure the 100 mm line. If it is not 100 mm on paper, do not use the template; correct print scaling or use the numeric station table.
Are layout points cut coordinates?
They are positions on an ideal arc measured from the chord baseline. Add machining or material allowances separately. The printer/template and table are planning aids, not machine instructions.