I have found that two dimensions are especially helpful in designing arches; the radius of the arch and the length of the boards that make up the arch. If you do a search on ‘Woodworking Formulas’ with the added keywords Spreadsheets, Arches, and such, you will hit on numerous sites that either have formulas like the one below, or java script calculators on their webpage, or even spreadsheets free to download. One of the best I’ve found was written by Paul Huntington, an architectural woodworker from Denver. His spreadsheet is very comprehensive and includes calculations for arcs, ring segmens, stairs, helix stairs, springback calculations for bent laminations, dualslope, crown calculations, bigarcs, ellipses, and proportions. Whew! That’s a lot of math. Here's a link to the Woodweb.com page where you can download the file.
To calculate the radius or length of an arc you only need two dimensions, the chord and the rise. The chord is any straight line that connects two points on the circumference except the diameter which is a special chord. (if you know the diameter you do not need to calculate the radius). The rise is the perpendicular distance from the middle of the chord to the circumference. The second diagram shows the formula for calculating the radius. You can also use the Dimension Tool in Sketchup to calculate the radius. On this project I used both the spreadsheet and Sketchup to confirm the calculation. The OD radius of the top rail is about 120”. Given a 3 1/2” width at the apex, the ID (inner diameter) radius is 116 1/2”. This means the ID radius of the cap rail is also 120” and the OD radius of the lower inside rail is 116 1/2”. Maybe I should just make the top rail and use that as a form for the cap and inner rails?
Notice that the 84” width of the cap rail in the Sketchup picture is not the actual length of the wood used to make the rail. That length, or the ‘Arc Length’ is closer to 86”. Just download the spreadsheet, it will make more sense than I do. Also check out Paul’s discussion on the Woodweb site.
-- tim hill www.newcalshop.com