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### Volume of a Dome

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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!
```

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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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Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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