how to calculate height between mid points on a chord and arc

• January 26th 2011, 09:50 PM
dsell
how to calculate height between mid points on a chord and arc

I am trying to build a shade house and need to get some steel bent. I need to provide the distance between the midpoint on the chord and the midpoint on the arc.

I have a chord length of 2.7mtrs and an arc length of 3.6mtrs. Goggling it gave me the formula c/s = sin(x)/x but I don't even know what that means. where c is the chord length and s is the arc length.

Sorry to sound dopey to all you maths gurus.
• January 27th 2011, 10:47 AM
Ithaka
Quote:

Originally Posted by dsell

I am trying to build a shade house and need to get some steel bent. I need to provide the distance between the midpoint on the chord and the midpoint on the arc.

I have a chord length of 2.7mtrs and an arc length of 3.6mtrs. Goggling it gave me the formula c/s = sin(x)/x but I don't even know what that means. where c is the chord length and s is the arc length.

Sorry to sound dopey to all you maths gurus.

arc length= $\theta*R$

chord length= $2*R*\sin(\frac{\theta}{2})$

where
$\theta$ =angle at the centre measured in radians and R= radius of the circle

$\theta*R=3.6$

and

$2*R*\sin(\frac{\theta}{2})=2.7$

If you divide the 2 equations above:

$\frac{\theta}{2*\sin(\frac{\theta}{2})}=\frac{3.6} {2.7}$

or, if we change the variable by:
$x=\frac{\theta}{2}$

then you get to:
$\frac{x}{sin x}=\frac{3.6}{2.7}$

or:

$\frac{sin x}{x}=\frac{2.7}{3.6}$

which is exactly your formula: sin(x)/x=c/s where c is length of chord (2.7)
and s is length of arc (3.6)

The equation can be re-written as:

$sin x=x*\frac{2.7}{3.6}$

So it all boils down to finding x = solving this equation which can be done by using numerical methods or a graphical method.

Knowing x will allow you find the angle at the centre $\theta$=2x
and then the radius of the circle R from:

$\theta*R=3.6$
$R=\frac{3.6}{\theta}$

Now, the distance you have been looking for is
$R-R*cos(\frac{\theta}{2})$
• January 27th 2011, 11:42 AM
stebko
Not formula, but calculation for the distance between the midpoint on the chord and the midpoint on the arc in your situation.
If a chord length of 2.7mtrs and an arc length of 3.6mtrs, this distance is 1.00 mtrs
• January 27th 2011, 12:17 PM
Ithaka
Quote:

Originally Posted by stebko
Not formula, but calculation for the distance between the midpoint on the chord and the midpoint on the arc in your situation.
If a chord length of 2.7mtrs and an arc length of 3.6mtrs, this distance is 1.00 mtrs

Ok: the first part was just a (long :) ) explanation why the formula works.

The calculation: by graphical method (see graph attached) the equation
sin x = x* 2.7/3.6 has solution x=1.28, $\theta=2.56$
so R=3.6/2.56=1.40625

therefore the distance = 1.40625-1.40625*cos 1.28 = 1.00 m, same answer as your (more practical) approach

The idea is that, once you are given the value of c (2.7) and (3.6 in this case), you can:

1. find x by solving (graphically) the equation sin x =x*c/s
2. find $\theta$ = 2*x
3. find $R=\frac{s}{\theta}$
4. find the distance you have been looking for = $R-R*cos{\frac{\theta}{2})$