I have a problem which I thought would be quite easy when I was out on site.
I have a curved bridge which I am assuming is part of a circle.
I measured the arc length AL and by pulling a string taught and dropping a ruler down measured the distance from the centre of the arc to the chord which I call h.
I thought that these two measurements must define an individual circle, and did not make an attempt to measure the chord length.
I am now trying to define the radius and angle etc. of the arc but cannot do this algebraically. I could probably attack this iteratively to find the particular solution using excel but I'd like to know if there is a direct equation to solve this.
I have been through a few sides of A4 trying to solve this and suspect I do not have enough information, so here is some of my equations I have written but cannot see how to combine them to get r the radius or theta alone.
r = radius
theta = half the included angle
AL = arc length
CL = chord length
h = Arc Height ( max distance from chord to Arc)
r*cos (theta) = r-h
CL = 2*r*sin( theta )
AL = r * theta
r^2 = (CL/2)^2 + (r-h)^2
What approach should I be taking?
(I have found an online calculator that has given me a result but I am frustrated with my own attempts to solve it directly)
Any help would be most appreciated.