Question 8055
 We can solve for the height directly without knowing the radius.
 If the arc length is a.
 Let the radius be r and the angle facing the arc be x (in radians),then 
 the arc length a = r x  ...(1)
 Set the chord be c (= AB, M the midpoint of AB, O center of the circle), then
 we have r sin x/2 = c/2...(2) [From at the right triangle AMO]
 
 Solve the system (1),(2)(2 equations in two variables)
 for r and angle x.
 Then we obtain the height
 r - r cos(x/2) = r(1 - cos x/2) [since OM = r cos x/2]
 (or by (2) use r cos(x/2) = r sqrt(1- sin^2 (x/2)] = r sqrt(1- c^2/4r^2])
  Hence, then the height = r(1 - cos x/2) = r(1-sqrt(1- c^2/4r^2]))

 Actually, it is not so easy to solve the system (1),(2). 
 Because it involves trig functions , so you may need to use Taylor's
 series or Newtons method to solve for r and x.

 More questions are welcome.

 Kenny