Area of an Ellipse Cut by a Chord

Date: 05/26/2000 at 14:21:46
From: Andy Jordan
Subject: Partial Area of an Ellipse

Hi,

I have been learning partial flow in pipes of different shapes in an 
upper level civil engineering class. To try to prove my math skills I 
have taken on the task of trying to figure out partial flow in an 
elliptical pipe. What I need to know is how you can calculate the area 
of part of the ellipse, if you know the major and minor axes, and you 
know the depth of the water, y. Note that the water has to sit on the 
bottom of the pipe due to gravity, so it's not just floating in the 
middle. I think if there is a formula, it will vary depending whether 
the major axis is horizontal or vertical. I know the area of an 
ellipse is pi*a*b, and I know that you can take the whole area of the 
ellipse and subtract the percentage of the pipe that isn't being used. 
But I don't know how to get that percentage or the area of the water.

Thanks for the help.

Sincerely,
Andy

Date: 05/26/2000 at 22:46:07
From: Doctor Peterson
Subject: Re: Partial Area of an Ellipse

Hi, Andy.

I'll just give you the basic idea behind the surprisingly simple 
solution, and let you see how well you can carry it out.

An ellipse is just a "stretched" (or "squashed") circle; that is, you 
can make an ellipse with semiaxes a and b by taking a circle with 
radius a and multiplying all y coordinates by b/a. When you stretch a 
figure that way, you multiply all areas by b/a; that's why the formula 
for area of a whole ellipse is:

     A = pi a^2 * b/a = pi a b

Now, if you just have a segment of an ellipse, you can undo this 
stretching and get a segment of a circle:

           ***********+*
       ****......|b....\****
      *..........|......\   *
     *-----------+-------\---*
      *.....a....|........\ *
       ****......|......***+
           *************

             *********
          ***....|....+**
        **.......|.....\ **
       *.........|a....\   *
      *..........|......\   *
     *...........|......\    *
     *-----------+-------\---*
     *.....a.....|........\  *
      *..........|........\ *
       *.........|.........+
        **.......|.......**
          ***....|....***
             *********

It will be easiest if the segment is cut parallel to one of the axes, 
which is likely in your problem according to the way you stated it, 
but I've shown the harder case, which can be done with just a little 
more trouble, in case you're allowing the pipe to be tilted. (You'll 
have to tilt my pictures, of course.)

So you can take the dimensions of your elliptical segment, find the 
corresponding segment of a circle and work out its area, then multiply 
by b/a.

You can either look up the formula for a segment of a circle in our 
FAQ (click on Formulas), or draw two radii and add the areas of the 
sector and the triangle that you form.

Let me know if you need more help.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/