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

1. ## 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.

2. 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
$R-R*cos(\frac{\theta}{2})$

3. 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

4. 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 = $R-R*cos{\frac{\theta}{2})$