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Suppose that in the following diagram, I am only given c and s, the lengths of a chord and its segment.

Is it possible to find the exact value of the radius using c and s?

Every formula I found involves using h (distance from centre of chord to centre of the arc) or theta but I am only given c and s. Is there a simple way to find h or theta then use it to find R?

enter image description here

user6005857
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1 Answers1

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$$S=R\theta$$

$$c=2R\sin\left(\frac{\theta}{2}\right)$$

Dividing we get,

$$\frac{c}{s} = \frac{\sin(t)}{t}$$

where $t=\theta/2$ and so $0< t < \pi/2$. Since $\sin(x)/x$ is 1-1 on $(0,\pi/2)$, a unique solution $t$ exists (assuming $2s/\pi < c < s$) and $R = \frac{s}{2t}$

Guy
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  • What if theta is unknown? I only know c and s. Can I find theta using c and s? – user6005857 May 29 '17 at 09:32
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    @user6005857. Then, you need to solve the equation. Have a look at https://math.stackexchange.com/questions/871516/how-could-i-improve-this-approximation – Claude Leibovici May 29 '17 at 09:34
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    This also could be of interest https://math.stackexchange.com/questions/2284522/how-to-find-the-maximum-height-perpendicular-to-arc-without-knowing-radius-of-th/2286314#2286314 – Claude Leibovici May 29 '17 at 09:37