I am working on the design of a curved wall in a building. The
horizontal length of the wall is exactly 34'-0".  The actual length of
the wall along the curve needs to be exactly 35'-0".  This curve is a
portion of a circle and is symmetrical.  What is the radius of the
circle and the degree of the arc needed to meet these two
requirements.  I would also like a general equation for solving this
problem with different spacings.  Thanks

Hi zkane,

A seemingly simple problem, this question is actually surprisingly
complex.  However a useful site that I found gives a guide as to how
to proceed:

Ask Dr. Math FAQ: Segments of Circles
http://mathforum.org/dr.math/faq/faq.circle.segment.html

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
http://mathforum.org/dr.math/faq/faq.circle.segment.html#1

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
formula

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]
= 0.4167

x(2) = 0.4167 - [sin(0.4167) - (0.9714*0.4167)]/[cos(0.4167) - 0.9714]
= 0.4174

(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
answer.

Thanks,

mcfly-ga :-)


Search strategy:

circle theorems
maths length chord
length chord arc

---

Request for Answer Clarification by zkane-ga on 03 Jan 2003 06:06 PST

Wow, now I don't feel so bad about not being able to solve this on my
own.  The numbers do work and I am happy with the answer.  My only
question is how to go about increasing the accuracy of the numbers. 
When I construct the arc with the given numbers I am getting errors
between .25" and .5".
Regardless I am really impressed with the answer and the whole -Google
Answers setup.

---

Clarification of Answer by mcfly-ga on 03 Jan 2003 10:21 PST

Hi zkane,

Thanks for the great rating!  The accuracy of the figures for radius
and subtended angle is limited by the number of iterations used when
applying the Newton Method.  In my answer I applied the formula twice
to get a precision of 2 decimal places.  To improve on a tolerance of
0.5 inches out of 41'-8'' requires precision of 4 decimal places.

Therefore substituting the value of x(2) into the formula 
x(n+1) = x(n) - (sin[x(n)]-kx(n))/(cos[x(n)]-k) 
and repeating until x(n+1) varies from x(n) by less than 0.00005 will
reduce the error even further.

Good luck in your wall building :)

mcfly-ga