June 8th, 2018, 01:31 AM  #1 
Newbie Joined: Jun 2018 From: Idaho Posts: 2 Thanks: 1  Need to calculate the length of an arc
Hello, Need to make an arc that is 16' wide at the base, and rises 12" in the centre. It'll be a perfectly curved arc. I need to know what the length of the beam for the arc should be. I looked up formulas online, but they all want the degree of the arc, but I won't have that until I build it. Would very much appreciate the help. 
June 8th, 2018, 01:58 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,879 Thanks: 1087 Math Focus: Elementary mathematics and beyond 
Using the information found here, calculate the radius of the circle that contains the arc. Can you proceed from there?

June 8th, 2018, 02:13 AM  #3 
Newbie Joined: Jun 2018 From: Idaho Posts: 2 Thanks: 1 
I believe so. I'll give it a shot. Thank you.

June 8th, 2018, 03:07 AM  #4 
Senior Member Joined: Jun 2015 From: England Posts: 887 Thanks: 266 
The full formula to find the radius is given in 1 in the sketch. I was working in inches for accuracy. This leads to a radius of 390" or 32' 6". But you probably don't have access to the centre of the circle. Here are two ways to proceed for setting out your arch. First you can use the exact formula and work horizontal distances up from the springings. Or you can use the approximate inverted formula and dip down from a string line stretched tightly across the gap. This is shown in 2. I calculate the maximum error in using the approximate formula is 0.2". If this is acceptable this is a much easier method to implement. 
July 18th, 2018, 08:26 PM  #5 
Newbie Joined: Jul 2018 From: tn. Posts: 4 Thanks: 0 
Saibaton: I don't think from the information given that this will not work. draw a horzonal line 16" long and from the center of that line, draw a vertical line 12" long. now draw an arc from the ends of the 16" line to the top of the 12" line, you have two arcs one on each side of the vertical line but the arcs will not be constant. don't think it can be done.

July 19th, 2018, 01:05 AM  #6  
Senior Member Joined: Jun 2015 From: England Posts: 887 Thanks: 266  Quote:
1) What does the double negative in the above quote mean? 2) What your objection is in detail.  
July 21st, 2018, 06:14 AM  #7 
Newbie Joined: Jul 2018 From: tn. Posts: 4 Thanks: 0 
the 16" line is a cord in a circle. So with that circle creating an arc from both ends of the 16" line no way that I see can produce that perfectly curved arc and be 12" from that 16' line. don't care much for formulas, just using common sense. Now you might create something with an ellipse, but he is asking for a perfectly curved arc. I may be missing something here.
Last edited by skipjack; July 21st, 2018 at 07:11 AM. 
July 21st, 2018, 07:22 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,722 Thanks: 1807 
What you're missing is that the width is 16', not 16".

July 21st, 2018, 07:59 AM  #9 
Senior Member Joined: Jun 2015 From: England Posts: 887 Thanks: 266 
Thank you Burt, for coming back to us. Yes, look carefully the OP is spanning a 16 foot gap with a timber slightly bent into the shape of an arch that has a rise of 1 foot. Note I worked in inches in my diagram. The diameter of that circular arch is 65 feet. (which was not supplied but had to be worked out. And yes it is possible to match the curvature exactly at a connection point of two separate curves if you are careful. Surveyors do this all the time. Finally there no reason why a circle with a chord (note the spelling, a cord is a measure of wooden logs) of length 16 inches cannot have a rise of 12 inches. It just means that the diameter, and therefore the centre, is above the springings, and the top is on the other side of the diameter from the chosen chord. This is because there are two chords of length 16 inches and the circle diameter is approximately 18 inches, as in the diagram. 
July 22nd, 2018, 05:50 PM  #10 
Newbie Joined: Jul 2018 From: tn. Posts: 4 Thanks: 0 
Thanks Studiot for the reply, and I thank skipjack for pointing out what I was missing 16 feet instead of 16 inches. I have a different approach to solve this question from Saibaton. I will put a sketch here if I knew how to load it, any suggestions on how I go about that let me know, thanks in advance, Burt.
Last edited by skipjack; July 23rd, 2018 at 12:14 AM. 

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arc, calculate, construction, geometry, length, math 
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