Volume of Liquid in a Cylinder

Date: 6/4/96 at 3:17:54
From: Bill Billman 
Subject: Special considerations to cylindrical measurements

Dr. Math,

I am experiencing a problem in understanding how to calculate the area 
under a chord of a circle.  Actually, the problem involves the amount 
of liquid in a cylinder that is laid in the horizontal plane.  
With a cylinder lying horizontally, I would like to know the partial 
liquid contents if I can determine the height of the liquid.  

Example:  Let's say there is a cylinder 100 inches in diameter and 200 
inches in length.  Knowing the diameter of the circle (100 inches) and 
the length (200 inches), is there a formula that will yield the amount 
of liquid when the tank has 2, 3, 4 .... standing inches in it?
If possible, please advise, and thank you for your time....

Bill Billman

Date: 6/4/96 at 8:14:14
From: Doctor Anthony
Subject: Re: Special considerations to cylindrical measurements

The area of cross-section of the liquid is found by calculating the 
area of the sector of the circle in which the liquid lies, and 
subtracting from this the area of the triangle formed by the surface 
of the liquid and the radii from the centre of the circle to the edges 
of the surface.  If O is the centre of the circle and AB the chord, 
then the required area of cross-section of liquid is given by:

  area of sector OAB - area of triangle OAB

Taking your figures with radius of circle 50 inches, and shall we say 
a depth of 10 inches of liquid, we first find the angle AOB.  This is 
done by finding half its value (say X) and then calculating the areas 
of the sector OAB and the triangle OAB.

    cos(X) = (50-10)/50 = 40/50  and so X = 36.87 degrees
                                       2X = 73.74 degrees
                                          = 1.287 radians
  
Formula for area of sector = (1/2)*(radius^2)*(angle in radians)

Area of sector = (1/2)*(50^2)*1.287 = 1608.75

Area of triangle = (1/2)*(OA*OB)*(sin(AOB))

                 = (1/2)*(50^2)*0.960001 = 1200.00125 

Subtracting these two areas we get 408.75 sq inches

The volume of liquid is then found by multiplying this area by the 
length of the cylinder (= 200 inches)

Required volume = 408.75*200 = 81749.75  cubic inches

-Doctor Anthony,  The Math Forum
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