Spanish Formulas to help do the figuring. February 28, 2004

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.

To find the radius of a curve segment, use the following:

2 x A x R = A squared + B squared

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:

R = 78.227272"

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.

Sweet. Sweet. Sweet. If I am right, the tan(theta)=B/(R-A) [that is the tan of the interior angle (really 1/2 of the angle formed by the two radii at the ends of the chord)] giving radians, multiplied by R gives the length of the curve! Yahooooo.

Radius of an arc knowing the chord length and arc height (same formula as above, just reduced differently):

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.

I have on the shelf here by the computer the CRC Standard Mathematical Tables (14th edition, 1965) that my brother gave me when I graduated from high school.

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.

Thanks ever so much to all. Makes a big difference. Now I don't have to guess that a 4x8 will do the job or not...

I've had a spreadsheet with many formulas available for download on this website for almost two years now:

Spreadsheet Calculation Program

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.

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Untitled Document

**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

4062Persimmon 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 - Accurate Miter Cuts on Very Large Beams
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