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Formula for Laying Out an Arc

Date: 09/27/2002 at 12:09:16
From: Carl Propson
Subject: Formula for laying out an arc using a known radius and 
distance from chord to the arc at 90 deg.

Hello Dr. Math,

I work as a layout carpenter and do not have the advantage of the 
newer transits. I am working on a building having a 317' radius with 
steel centers along it, and will not be able to swing this radius with 
a measuring tape. If I could figure the layout from a chord it would 
be a great help. There must be a formula for this.

The radius is known, the distance from the center of the chord at 
90 deg. to the arc is known, the distance from the center of the chord 
along the chord to a point is known - call it A. What would be the 
distance from A to the point where it intersects the arc at 90 deg. 
off the chord.            

Your help would be greatly appreciated.

Thank you very much,
Carl


Date: 09/27/2002 at 15:34:54
From: Doctor Rick
Subject: Re: Formula for laying out an arc using a known radius and 
distance from chord to the arc at 90 deg.

Hi, Carl.

I think I understand what you're saying, and it's not hard to work out 
using coordinate geometry. We can lay out our coordinate system so 
that the chord is along the x-axis with the center of the chord at the 
origin (0,0). If the radius is r and the perpendicular distance from 
the center of the chord to the arc is h, then the center of the circle 
is at (0, h-r), so the equation for the circle is

  x^2 + (y-(h-r))^2 = r^2

We can solve for y in terms of x:

  y = sqrt(r^2 - x^2) - r + h

This is what you're looking for: x is the distance along the chord 
from the center of the chord to your point A, and y is the 
perpendicular distance from point A to the arc.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 09/29/2002 at 19:59:15
From: Carl Propson
Subject: Formula for laying out an arc using a known radius and 
distance from chord to the arc at 90 deg.

Hello Dr Rick,

Thank you very much for all your help! Knowing that formula will 
make the job much easier. 

I do however have another question: is there a formula to determine 
the angle at the point where the line from point A running 90 deg. 
from the chord intersects the arc ? 

I must set the anchor bolts along the arc using this angle.

Thanks again, 
Carl


Date: 09/30/2002 at 08:26:55
From: Doctor Rick
Subject: Re: Formula for laying out an arc using a known radius and 
distance from chord to the arc at 90 deg.

Hi, Carl.

Yes, the angle can be found using right-triangle trigonometry. We have 
a right triangle with one vertex at the center of the circle, C; 
another where the perpendicular line from the chord at A intersects 
the circle, which I'll call B; and the third where a line from the 
second vertex running parallel to the chord intersects the line 
through the circle center and the chord center - I'll call this D. 
The hypotenuse, CB, is the radius of the circle, r. The side opposite 
the center of the circle, BD, has length x. The sine of angle DCB is 
the ratio x/r. This angle is the same as the angle between the chord 
and the tangent to the circle at B (since CB is perpendicular to the 
tangent, and BD is perpendicular to the chord). Thus the angle you 
seek is arcsin(x/r).

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 10/26/2002 at 10:01:17
From: Carl Propson
Subject: Thank you (Formula for laying out an arc using a known radius 
and distance from chord to the arc at 90 deg.)

Thank you very much for your help; I am using these formlas daily in 
laying out the building.

Thanks again, 
Carl
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
High School Practical Geometry

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