Volume of Inscribed Cylinder

Date: 08/14/97 at 12:53:14
From: Courtney Denton
Subject: The volume of an inscribed cylinder

A cylinder of height h is inscribed in a sphere of radius q. (The 
edges of the top and base of the cylinder are touching the sphere.)  
Find an expression for the volume of the cylinder.

The answer to this question is (pi)(q^2)(h) - (xh^3)/(4)  I understand
the first part of this answer, but I don't understand where they got 
the second part (after the -).  Help would be greatly appreciated.

Thanks, Courtney

Date: 08/18/97 at 11:40:01
From: Doctor Marko
Subject: Re: The volume of an inscribed cylinder

Hi Courtney,

I want you to imagine (or even better draw on a piece of paper) a 
circle with a rectangle inscribed, such that the corners of the 
rectangle touch the circle. You can label the vertical side of the 
rectangle h and the horizontal side d.  

Now imagine that this picture rotates around a line parallel to h 
and through the center of the circle. You will get exactly the 
problem that you are asking about - a cylinder inscribed in a sphere.

We know that the volume of any cylinder is (pi)(radius^2)(height), and 
so those are the things that we need to figure out about our cylinder, 
in terms of things that we know, namely h, and q - the radius of the 
sphere. At this point you notice that the radius of the cylinder is 
NOT q and that it is the only thing you need to figure out in order to 
answer your question. The radius, you notice, is exactly d/2, so we 
only need to figure out d.

Now go back to your picture and draw a radius q such that it touches 
the circle at one of the corners of the rectangle, and another that 
touches the opposite corner. Now you have a right triangle and 
therefore can use the Pythagorean Theorem: 

  (2q)^2 = h^2+d^2.  
 
So     V = (pi)(d/2)^2 (h) = (pi)(d^2)(h)/4   and from the Theorem 

     d^2 = 4(q^2)-h^2.

Then   V = (pi)[4(q^2)-(h^2)](h)/4 = (pi)(q^2)(h)-(pi)(h^3)/4

I hope that this has helped.

-Doctor Marko,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/