a student before me designed 3 curves and reported the dimensions as Radius of curvature (radius of the circle) and the distance h (see this diagram Circular Segment -- from Wolfram MathWorld). i have to design 2 more curves for a given h (the arc length S is constant at 48in.) how do i go about calculating c and R based solely on S and h?
i've found a method but it gives me somewhat high errors (for increasing h) based on what the previous student reported as the R. the formula i used was R = a*b*c / [4*sqrt(S*(S-a)*(S-b)*(S-c))]
where S is the arc length and is also given by S = a+b+c / 2. BUT a = c and the triangle is equilateral so c can be found to be c = S^2-h^2 / S and subsequently, a can be found as well. is there a better / more accurate way to do this? or am i doing it correctly?
previous student's numbers
h = 2 / R = 143.67
h = 4 / R = 71.325
h = 6 / R = 46.965
my numbers and errors (based on his, granted i don't know if he even did it correctly).
h = 2 / R = 144.5 / error = .578%
h = 4 / R = 73 / error = 2.35%
h = 6 / R = 49.51 / error = 5.422%