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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
circles').

Segments of Circles - Dr. Math FAQ
http://mathforum.org/dr.math/faq/faq.circle.segment.html

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.

Michelle
```

```
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 = ----------
8h

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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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