A seemingly simple problem, this question is actually surprisingly
complex. However a useful site that I found gives a guide as to how
Ask Dr. Math FAQ: Segments of Circles
Using the parameters defined by this site, we know the arc length 's'
to be 35, chord length 'c' to be 34, and we wish to determine the
circle radius 'r' and the angle subtended by the arc, 'theta'.
Using rules of right-angled triangles, we can say that
sin(theta/2) = c / 2r (1)
If we initially measure angle theta in radians (converting into
degrees can be done later), then
theta = s / r and therefore r = s / theta (2)
Substituting equation (2) into (1) we get
sin(theta/2) = (c*theta)/2s
Renaming theta/2 to be 'x' and rearranging gives the formula quoted at
c / s = sin(x) / x (3)
We know that theta = 2x and r = s / theta so we just need to
solve equation 3.
Unfortunately the solution to sin(x)/x comprises of an infinite
series but an accurate approximation can be achieved using Newton's
Method. This uses the repeated application (interation) of the
x(n+1) = x(n) - (sin[x(n)]-kx(n))/(cos[x(n)]-k)
where k = c /s .
This may seem rather overwhelmingly complex, but working with our
numeric example (c=34, s=35) simplifies things somewhat.
k = 34/35 = 0.9714
A suggested starting value for the iteration is x(0) = sqrt(6-6k) =
0.4140, then applying the formula repeatedly gives:
x(1) = 0.4140 - [sin(0.4140) - (0.9714*0.4140)]/[cos(0.4140) - 0.9714]
x(2) = 0.4167 - [sin(0.4167) - (0.9714*0.4167)]/[cos(0.4167) - 0.9714]
(NOTE: These calculations must be done with a calculator in 'rad'
mode, not 'deg')
This should be enough for us to say with a high level of certainty
that x = 0.42 to two decimal places.
Therefore, in radians, angle theta = 2*0.42 = 0.84 and radius r =
35/0.84 = 41.7' = 41' 8''
To convert theta into degrees we can multiply by 180/pi to give theta
= 48.1 degrees.
In answer to your question then, an arc of 48.1 degrees of a circle of
radius 41'-8'' will give the wall you describe. The above process can
of course be applied for any values of 's' and 'c'.
I hope this answer has been both interesting and informative. If you
would like any clarifications please request them before rating my
maths length chord
length chord arc