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Area under Arc of Circle

Date: 02/27/2001 at 18:29:32
From: Michelle Drapeau
Subject: Area under arc of circle

Dear Dr. Math,

I would like to calculate the area delimited by the arc of a circle 
and the chord of that arc when (this is the important part) all I 
have is the length of the chord and the length of a line perpendicular 
to the chord and running from the middle of the chord to the edge of 
the circle (noted as letter h in the archive section 'segments of 

  Segments of Circles - Dr. Math FAQ   

All formulas I have found, including the ones found on the above FAQ 
page, require either theta (or alpha) or the radius, I have neither. 
Thank you for your help,

Date: 02/27/2001 at 23:19:04
From: Doctor Peterson
Subject: Re: Area under arc of circle

You have h and c. The FAQ gives these formulas:

K = r^2[theta-sin(theta)]/2 = r^2 arccos([r-h]/r) - (r-h)sqrt(2rh-h^2) 
  = r^2 arccos(d/r) - d sqrt(r^2-d^2)

What you missed is that the FAQ also tells you how to find c given h 
and r:

    c = ... = 2 sqrt[h(2r-h)]

You can solve this to find r in terms of c and h, the values you have:

        c^2 + 4h^2
    r = ----------

Now plug that into the equation for K in terms of r and h, and you'll 
have what you need. It will be horribly ugly, but it works!

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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