Volume of a DomeDate: 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/ |
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