I'm looking for a general formula for calculating the length of curve given the chord length at the endpoints, the height to the curve from the mid-point of the curve and the radius of the curve, and how to get one variable given the others. I have several projects where I know the length of the chord and the height to the curve, but I need to know what the length of the curve would be. Also the radius of the curve.
I've always found the little palm-sized "Engineers Handbook" useful. It's a soft cover that's about 3/4" thick and it's usually found by the register in HW stores for about $9 or $10. In addition to pretty much all the trig formulas you'd ever need, it's got electrical info, specific weights of things, conversion tables, etc. - in other words, a lot of stuff.
Chord length of the curve segment is 80", then B = 40" and the height of the curve line from the chord line (a straight line from one endpoint to the other) is A at 11". Plug them in to get:
2 x 11 x R = 121 + 1600
Work that down to:
22 x R = 1721
Reduce that by dividing the left and right sides both by 22, and you get:
A whole lot harder to type than it is to do. Write up a sample with the numbers all plugged in and it's easy to use. Turn it around to work backwards - say if you have the radius and height, need to know the length.
R= (((A/2)^2) + (B^2)) / 2B
(R = radius) (A = Chord Length) (B = Arc Height)
I say this as:
A over 2 squared plus B squared divided by 2B.
Arc length when you know the chord length and the radius:
Angle in Radians = 2 * ASIN((Chord Length) / (2 * Arc Radius))
Arc Length = Arc Radius * Angle in Radians
Arc length equals arc radius times the angle in radians.
The angle in radians equals 2 times ASIN of the chord length divided by 2 times the arc radius.
Arc height when you know the chord length and radius:
B = R - sqrt( R^2 - (A/2)^2 )
(B = Arc Height) (R = Radius) (A = Chord Length)
B equals R minus the square root of R squared minus A over 2 squared.
Besides all those useful logarithm, trig, exponential, hyperbolic trig, ... tables are sections on geometry, trig, etc. formulas that I need when designing and building new pieces. No joke. I even use my slide rule in the shop so I don't have to worry about keeping batteries in a calculator.
The Excel file has a whole lot more formulas and drawings than the original works spreadsheet. I wanted a quick, accurate way to calculate arcs for work I did. And I expanded on those ideas and made other useful formulas for woodworking. Best of all this is a free download for everyone.