Question
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.
Forum Responses
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.
For example:
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.
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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
Said as:
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.
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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)
Said as:
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.
Common Lumber Name | A | B | C |
Hardwoods | |||
Alder, Red | 9.9 | 19.2 | 2506 |
Apple | 10.9 | 31.7 | 4132 |
Ash, Black | 9.3 | 23.4 | 4132 |
Ash, Green | 14.3 | 27.6 | 3590 |
Aspen, Bigtooth | 10.3 | 18.7 | 2439 |
Aspen, Quaking | 10.3 | 18.2 | 2373 |
Basswood | 6.2 | 16.6 | 2174 |
Beech, American | 8.9 | 29.1 | 3793 |
Birch, Paper | 8.8 | 25.0 | 3260 |
Birch, Sweet | 11.9 | 31.2 | 4065 |
Birch, Yellow | 9.2 | 28.6 | 3723 |
Buckeye | 8.9 | 17.2 | 2235 |
Butternut | 11.3 | 18.7 | 2440 |
Cherry | 13.8 | 24.4 | 3184 |
Chesnut, American | 11.6 | 20.8 | 2708 |
Cottonwood | 8.5 | 16.1 | 2102 |
Dogwood | 6.8 | 33.3 | 4331 |
Elm, American | 10.2 | 23.9 | 3116 |
Elm, Rock | 12.2 | 29.6 | 3860 |
Elm, slippery | 11.5 | 25.0 | 3251 |
Hackberry | 11.8 | 25.5 | 3319 |
Hickory, Bitternut (Pecan) | 14.7 | 31.2 | 4062 |
Hickory (True) | |||
Hickory, Mockernut | 9.1 | 33.3 | 4332 |
Hickory, Pignut | 9.3 | 34.3 | 4332 |
Hickory, Shagbark | 10.9 | 33.3 | 4333 |
Hickory, Shellbark | 6.6 | 32.2 | 4195 |
Holly, American | 8.3 | 26.0 | 3387 |
Hophornbeam, Eastern | 7.9 | 32.8 | 4266 |
Laurel, California | 15.1 | 26.5 | 3456 |
Locust, Black | 21.2 | 34.3 | 4470 |
Madrone, Pacific | 7.8 | 30.2 | 3925 |
Maple (Soft) | |||
Maple, Bigleaf | 12.8 | 22.9 | 2980 |
Maple, Red | 13.1 | 25.5 | 3318 |
Maple, Silver | 12.4 | 22.9 | 2981 |
Maple (Hard) | |||
Maple, Black | 12.3 | 27.0 | 3523 |
Maple, Sugar | 12.3 | 29.1 | 3793 |
Oak (Red) | |||
Oak, Black | 11.7 | 29.1 | 3792 |
Oak, California black | 16.4 | 26.5 | 3455 |
Oak, Laurel | 6.3 | 29.1 | 3791 |
Oak, Northern red | 13.6 | 29.1 | 3793 |
Oak, Pin | 13.0 | 30.2 | 3928 |
Oak, Scarlet | 13.2 | 31.2 | 4065 |
Oak, Southern red | 9.6 | 27.0 | 3520 |
Oak, Water | 10.4 | 29.1 | 3793 |
Oak, Willow | 6.4 | 29.1 | 3790 |
Oak (White) | |||
Oak, Bur | 15.4 | 30.2 | 3928 |
Oak, Chestnut | 10.1 | 29.6 | 3858 |
Oak, Live | 17.5 | 41.6 | 5417 |
Oak, Overcup | 10.7 | 29.6 | 3860 |
Oak, Post | 11.0 | 31.2 | 4063 |
Oak, Swamp chestnut | 10.7 | 31.2 | 4063 |
Oak, White | 10.8 | 31.2 | 4062 |
Persimmon | 7.0 | 33.3 | 4332 |
Sweetgum | 8.9 | 23.9 | 3115 |
Sycamore | 10.7 | 23.9 | 3115 |
Tanoak | 9.0 | 30.2 | 3926 |
Tupelo, Black | 10.4 | 23.9 | 3116 |
Tupelo, Water | 12.4 | 23.9 | 3115 |
Walnut | 13.4 | 26.5 | 3454 |
Willow, Black | 8.6 | 18.7 | 2438 |
Yellow-poplar | 10.6 | 20.8 | 2708 |
Common Lumber Name | A | B | C |
Softwoods | |||
Baldcypress | 13.2 | 21.9 | 2844 |
Cedar, Alaska | 14.4 | 21.9 | 2844 |
Cedar, Atlantic white | 10.9 | 16.1 | 2100 |
Cedar, eastern red | 16.4 | 22.9 | 2981 |
Cedar, Incense | 13.1 | 18.2 | 2371 |
Cedar, Northern white | 11.1 | 15.1 | 1964 |
Cedar, Port-Orford | 12.6 | 20.2 | 2641 |
Cedar, Western red | 12.2 | 16.1 | 2100 |
Douglas-fir, Coast type | 12.3 | 23.4 | 3049 |
Douglas-fir, Interior west | 13.2 | 23.9 | 3116 |
Douglas-fir, Interior north | 14.0 | 23.4 | 3048 |
Fir, Balsam | 9.9 | 17.2 | 2236 |
Fir, California red | 10.6 | 18.7 | 2437 |
Fir, Grand | 10.7 | 18.2 | 2371 |
Fir, Noble | 10.1 | 19.2 | 2507 |
Fir, Pacific silver | 10.4 | 20.8 | 2711 |
Fir, Subalpine | 10.5 | 16.1 | 2101 |
Fir, White | 12.2 | 19.2 | 2506 |
Hemlock, Eastern | 12.6 | 19.8 | 2573 |
Hemlock, Western | 11.5 | 21.8 | 2847 |
Larch, Western | 11.3 | 25.0 | 3251 |
Pine, Eastern white | 12.3 | 17.7 | 2303 |
Pine, Lodgepole | 11.5 | 19.8 | 2576 |
Pine, Ponderosa | 12.6 | 19.8 | 2573 |
Pine, Red | 12.2 | 21.3 | 2777 |
Southern yellow group | |||
Pine, Loblolly | 12.9 | 24.4 | 3183 |
Pine, Longleaf | 15.0 | 28.1 | 3658 |
Pine, Shortleaf | 12.9 | 24.4 | 3183 |
Pine, Sugar | 12.6 | 17.7 | 2302 |
Pine, Western white | 10.0 | 18.2 | 2370 |
Redwood, Old growth | 14.9 | 19.8 | 2573 |
Redwood, Second growth | 13.2 | 17.7 | 2302 |
Spruce, Black | 11.3 | 19.8 | 2575 |
Spruce, Engelmann | 10.0 | 17.2 | 2234 |
Spruce, Red | 10.6 | 19.2 | 2506 |
Spruce, Sitka | 10.8 | 19.2 | 2506 |
Tamarack | 12.0 | 25.5 | 3318 |