Volume of a Dome

Date: 03/09/99 at 14:57:24
From: Ethan street
Subject: Volume of a Dome

Is there a universal formula for the volume of (how I refer to it as) 
a dome? For example, if I was given a sphere and I took a plane that 
intersected the sphere dividing it into two sections; how would I go 
about finding the volume of one of the sections? I am pretty sure it 
involves calculus, but I am not positive. I would be extremely 
grateful for such a formula. I do not even know where to start!

Date: 03/09/99 at 15:50:01
From: Doctor Wilkinson
Subject: Re: Volume of a Dome

Let us assume that the sphere has radius r and that you divide it by a 
plane at height h above the equator. (I will assume h is positive. You 
can handle the case where the 'dome' is more than half the sphere 
easily enough once you have done the other case.) Then think of 
slicing the sphere with a bunch of planes parallel to the given plane 
but above it, so that the whole dome-shaped volume is divided into very 
thin slices. The volume of a slice of thickness d is then very close to 
the area of the circular cross-section at the bottom of the slice times 
the thickness d. Adding up the volumes of all the slices gives you an 
approximation to the volume of the dome. If you make d smaller and 
smaller, the approximation approaches the true volume, and this is 
given by the integral from h to r of 

   pi (r^2 - x^2) dx

which is just

   pi r^2 (r - h) - pi (r^3/3 - h^3/3)

For more about sphere formulas, see the Dr. Math FAQ:

  http://mathforum.org/dr.math/faq/formulas/faq.sphere.html   

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