# Arc Length given Chord Length & Height

#### mms

s = Arcsin(c / (h + c^2 / 4h)) X (h + c^2 / 4h)

given c = 163' & h = 7'

I do

s = Arcsin (163 / (7 + 163^2 / 4 X 7)) X (7 + 163^2 / 4 X 7)
s = Arcsin (163 / 955.89) X (955.89)
s = Arcsin (0.1705) X (955.89)
s = 9.818 X 955.89
s = 9,385'

This is obviously not correct.

What am I doing wrong?

#### studiot

Can you not do this one from first principles more easily than finding some long-winded formula?

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#### mms

Now I guess my question is one of curiosity on my part.

Is the published formula correct?
If it is correct what am I doing wrong, as I get an incorrect answer when I solve for arc length using it.

#### skipjack

Forum Staff
What am I doing wrong?
You evaluated the arcsin in degrees. You should have used radians.

By the way, is c exactly 163'?

#### mms

Thanks skipjack

Redoing:

s = Arcsin (163 / (7 + 163^2 / 4 X 7)) X (7 + 163^2 / 4 X 7)
s = Arcsin (163 / 955.89) X (955.89)
s = Arcsin (0.1705) X (955.89)
s = Arcsin (0.1705 X pi / 180) X (955.89)
s = Arcsin (0.002976 rad) X (955.89)
s = 0.1705 X 955.89
s = 163.00024063'

Still don't get 163.83' as does studiot

By the way, is c exactly 163'?
Engineer draws a curved member with chord length dimensioned at 163'-0" and no more information.
7' is best I can measure from engineering drawing.

#### skipjack

Forum Staff
and 0.1713586 Ã— 955.893 = 163.8 approximately.

#### mms

s = Arcsin (163 / (7 + 163^2 / 4 X 7)) X (7 + 163^2 / 4 X 7)
s = Arcsin (163 / 955.89) X (955.89)
s = Arcsin (0.1705)radians X (955.89)
s = 9.8169radians X pi /180 X 955.89
s = 0.17134degrees X 955.89
s = 163.78' If I carry full decimal places in the above steps I get
s = 163.800456979'

Thanks skipjack

#### studiot

Since this thread is obviously still live here are some useful formulae.

These are much better than the clumsy arcsin one from Wiki.

First Engineers used to use versed sine (versine) tables before calculators.

This easily gives the maximum rise from the chord if the radius and angle subtended to the centre are known.

A great many more offsets from the chord than just the centre one will be needed.

These can be calculated as in the first formula, after calculating the max offset.

An alternative are the offsets from the tangent (or any sideways displaced straight line) as shown in the second diagram. #### Attachments

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#### mms

Thanks studiot

I can't decipher the "?" below

h = ho? - { R - sqrt(R^2 - x^2) }

#### studiot

c/2 is the half chord length.

ho is the maximmum height of the arc above the chord and is at the centre of the chord and arc.

Distance x is measured from the centre.

h is the height of the arc above any point at distance x from the centre and is equal to ho minus the expression (square root) in brackets.

It arises thus:

The arc is symmetrical about the centre so if you divide the chord into an even number of sections as in the diagram you calculate two offsets at a time, one each side of the centre.

The geometry of the formula is shown in the diagram.
What you are effectively doing is repeatedly using the centre height formula across a smaller and smaller chord as shown by the dashed line and subtracting it from the main centre height.

Before computers drawing offices had what were known as 'railway curves' which were shaped pieces of wood or plastic formed to arcs of standard curvature, for drawing arcs of very large circle.

Surveyors used the formulae above for setting out such large curves.

I would hate all this old knowledge to be lost just because we can plot it out by computer these days.

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