SOLUTION: Is it possible to calculate the height from a given chord to a given arc, given that the only information available is the length of both the chord and the arc, I have been trying

Question 8055: Is it possible to calculate the height from a given chord to a given arc, given that the only information available is the length of both the chord and the arc, I have been trying to calculate this for some time but it is in vain?

Found 2 solutions by Earlsdon, khwang:
Answer by Earlsdon(6294) (Show Source): You can put this solution on YOUR website!
I would say that it is possible, if you also have the radius of the circle or the angle subtended by the arc.
Here's one approach using the radius: A picture might help.
Draw a circle of radius r.
Draw a chord of length c.
Draw two radii from the circle center to the ends of the chord.
Draw the radius from the circle center to the circumference, bisecting the chord.
Now, label the segment of the radius last drawn from the center to the chord as x.
Label the remaining segment, the height you are looking for, as h.
Now, if drawn correctly, your diagram should show an isosceles triangle formed by two radii and the chord. The isosceles triangle is bisected into two right-triangles with the radius (r) as their hypotenuses.
Using the Pythagorean theorem, you can find the length of segment x.

Once you have found the length of segment x, the length of h is simply r - x.
So you have:

You might see this written this way:

Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
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
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