Question

flash582
I have a part of a circle with a cord length of 8 inches. The area of the circle segment is 13.08 inches. How do I find the arc length of the portion of the circle?

Answer 1

The radius might be helpful. Find that.

Answer 2

Okay since this has gone 10 hours without a reply I just thought I'd try an outright guess.
radius = 4.6222
central angle = 119.85°

Answer 3

Pretty sure that was a reply. How did you get that radius? Please show your work.

Answer 4

You can solve this numerically. One way is find an equation for the radius, and find the root using, for example, newton-raphson

Area of a segment is
\[ A= \frac{r^2}{2}\left(\theta -\sin\theta\right) \]
Given that the chord is L, we require
\[ r \sin\left(\frac{\theta}{2}\right)= \frac{L}{2} \]
and
\[ r^2 = \frac{L^2}{4\sin^2\left(\frac{\theta}{2}\right)} \]
we get
\[ A= \frac{L^2}{8\sin^2\left(\frac{\theta}{2}\right)} \left(\theta -\sin\theta\right) \]
and
\[ f(\theta)= \frac{L^2}{4\left(1-\cos\theta\right)} \left(\theta -\sin\theta\right) -A\]

with an initial guess for θ, iterate using
\[ \theta_{n+1} = \theta_n - \frac{ f(\theta)}{ f'(\theta)}\]
where \( f'(\theta)\) is the derivative of f(θ) with respect to θ
once you have θ, you can find r, and the arc length s=rθ

Answer 5

you get
r = 4.622385207801005
th = 2.091713008867705 radians
arc len s = r*th= 9.668703271155012

for newton-raphson see
http://www.sosmath.com/calculus/diff/der07/der07.html

Answer 6

tkhunny
Since this had gone 10 hours without a reply I figured I'd take my own "stab" at it. I used an online circle calculator and used the "brute force" method of inputting various amounts for radius, central angle, etc and see how the answers compared to segment height area, etc. I think it's astounding that using such an unscientific method, I got the answer to 5 significant figures.