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

  1. #1
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    Question how to calculate height between mid points on a chord and arc

    Hi I am absolutely useless at math could someone please help me with this.

    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.
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  2. #2
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    Quote Originally Posted by dsell View Post
    Hi I am absolutely useless at math could someone please help me with this.

    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

    using your figures, you get:

    \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})
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  3. #3
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    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
    Attached Thumbnails Attached Thumbnails how to calculate height between mid points on a chord and arc-0002.jpg  
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  4. #4
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    Quote Originally Posted by stebko View Post
    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})
    Attached Thumbnails Attached Thumbnails how to calculate height between mid points on a chord and arc-graph.jpg  
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