Associated Topics || Dr. Math Home || Search Dr. Math

### 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?

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

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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search