Radius from Arc

**askmemath** · April 26th 2006, 06:19 AM

Please Help

If the arc length s and the distance h from the centre point of the associated chord to the arc is known, how do you calculate the radius?

Thank You

**CaptainBlack** · April 26th 2006, 09:16 AM

Originally Posted by askmemath

Please Help

If the arc length s and the distance h from the centre point of the associated chord to the arc is known, how do you calculate the radius?

Thank You

I don't think you will find a closed form solution for this problem - though
I am prepared to be proven wrong.

RonL

**ThePerfectHacker** · April 28th 2006, 01:12 PM

Here is a picture to help you out.
By the theorem of chord in a circle we have that,
$n^2=r(2r-h)$
But, $n=r\sin(s/2r)$
Thus, we have,
$r^2\sin(s/2r)=r(2r-h)$ [/tex]
Thus,
$r\sin(s/2r)=2r-h$
----------------------
If you know calculus
Thus, you need to find the zero's of the function,
$f(x)=x\sin(s/2x)-2x+h$
Use Newton's method

**CaptainBlack** · April 28th 2006, 01:18 PM

Originally Posted by ThePerfectHacker

Here is a picture to help you out.
By the theorem of chord in a circle we have that,
$n^2=r(2r-h)$

Just as well that I checked my note on this problem

The intersection chord theorem would give in this case:

$<br /> n^2=h(2r-h)<br />$

surly?

RonL

**ThePerfectHacker** · April 29th 2006, 06:49 PM

Originally Posted by CaptainBlack

Just as well that I checked my note on this problem

The intersection chord theorem would give in this case:

$<br /> n^2=h(2r-h)<br />$

surly?

RonL

One thing that confuses me is how an immortal (me) makes such a mistake

**earboth** · April 29th 2006, 09:21 PM

Originally Posted by ThePerfectHacker

Here is a picture to help you out.
By the theorem of chord in a circle we have that,
$n^2=r(2r-h)$
But, $n=r\sin(s/2r)$
Thus, we have,
$r^2\sin(s/2r)=r(2r-h)$
...

Hello,

I'm a little bit confused: When you plug in the value of n, shouldn't be there a squared sine value too?:

$r^2\left(\sin(s/2r)\right)^2=r(2r-h)$

Greetings

EB

**CaptainBlack** · April 29th 2006, 09:42 PM

Originally Posted by earboth

Hello,

I'm a little bit confused: When you plug in the value of n, shouldn't be there a squared sine value too?:

$r^2\left(\sin(s/2r)\right)^2=r(2r-h)$

Greetings

EB

There should be, now how did I miss that (its in my notes)

RonL

**earboth** · April 30th 2006, 04:36 AM

Originally Posted by ThePerfectHacker

One thing that confuses me is how an immortal (me) makes such a mistake

Oh, PLEASE!

don't gamble with your rule nr. 5

EB

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