Calculating Radians/Chords

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Formulas to help do the figuring. February 28, 2004

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.

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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.

From the original questioner:
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.


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.


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.

From the original questioner:
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.