My Math Forum  

Go Back   My Math Forum > Math Forums > Math

Math General Math Forum - For general math related discussion and news


Reply
 
LinkBack Thread Tools Display Modes
July 29th, 2015, 07:16 AM   #1
mms
Member
 
Joined: Nov 2009
From: Hamilton, Ontario

Posts: 30
Thanks: 0

Arc Length given Chord Length & Height

I got a formula for arc length from this page https://en.wikipedia.org/wiki/Circular_segment

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?
mms is offline  
 
July 29th, 2015, 11:02 AM   #2
Senior Member
 
Joined: Jun 2015
From: England

Posts: 915
Thanks: 271

Can you not do this one from first principles more easily than finding some long-winded formula?
Attached Images
File Type: jpg arc1.jpg (71.5 KB, 13 views)

Last edited by skipjack; July 29th, 2015 at 01:11 PM.
studiot is offline  
July 29th, 2015, 12:17 PM   #3
mms
Member
 
Joined: Nov 2009
From: Hamilton, Ontario

Posts: 30
Thanks: 0

Thanks studiot for your solution.

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.
mms is offline  
July 29th, 2015, 01:41 PM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 20,833
Thanks: 2161

Quote:
Originally Posted by mms View Post
What am I doing wrong?
You evaluated the arcsin in degrees. You should have used radians.

By the way, is c exactly 163'?
skipjack is online now  
July 30th, 2015, 05:26 AM   #5
mms
Member
 
Joined: Nov 2009
From: Hamilton, Ontario

Posts: 30
Thanks: 0

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

Quote:
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.
mms is offline  
July 30th, 2015, 07:29 AM   #6
Global Moderator
 
Joined: Dec 2006

Posts: 20,833
Thanks: 2161

arcsin(0.1705212) = 0.1713586 (radians) approximately,
and 0.1713586 × 955.893 = 163.8 approximately.
skipjack is online now  
July 30th, 2015, 08:49 AM   #7
mms
Member
 
Joined: Nov 2009
From: Hamilton, Ontario

Posts: 30
Thanks: 0

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
mms is offline  
July 30th, 2015, 09:49 AM   #8
Senior Member
 
Joined: Jun 2015
From: England

Posts: 915
Thanks: 271

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.

Attached Images
File Type: jpg circle2.jpg (42.6 KB, 5 views)
studiot is offline  
July 30th, 2015, 11:01 AM   #9
mms
Member
 
Joined: Nov 2009
From: Hamilton, Ontario

Posts: 30
Thanks: 0

Thanks studiot

Your additional formulae would be very useful to me.

I can't decipher the "?" below

h = ho? - { R - sqrt(R^2 - x^2) }
mms is offline  
July 30th, 2015, 01:08 PM   #10
Senior Member
 
Joined: Jun 2015
From: England

Posts: 915
Thanks: 271

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.
Attached Images
File Type: jpg circle3.jpg (32.6 KB, 1 views)

Last edited by studiot; July 30th, 2015 at 01:14 PM.
studiot is offline  
Reply

  My Math Forum > Math Forums > Math

Tags
arc, chord, height, length



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Base length of trapezoid given area, height, and non-parallel side lengths caters Geometry 3 March 11th, 2015 08:58 AM
Looking for a formula to calculate a chord length rainycat Algebra 2 July 23rd, 2013 11:50 PM
Help with the proper Calculation for Chord length tiehen Algebra 2 October 28th, 2012 05:46 AM
how do I calculate radius of a circle, from chord length? ,, boki-san Algebra 1 February 13th, 2011 10:34 AM
Chord length Aakash0815 Algebra 2 August 6th, 2009 05:00 PM





Copyright © 2019 My Math Forum. All rights reserved.