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### Volume equations for a sphere and pyramid

```
Date: 6/10/96 at 18:59:48
From: Ron C. Maggiano
Subject: How do you get the volume equations for a sphere and pyramid?

Hey Dr. Math,

We just learned the formulas for volume of a sphere (one third of four
times the side to the cube), and volume of a pyramid (one third of base
times the height,) but our geometry teacher wouldn't tell me how to get
them. Can you help?

Thanks!
```

```
Date: 7/1/96 at 17:0:16
From: Doctor Paul
Subject: Re: How do you get the volume equations for a sphere and
pyramid?

The Volume of a Sphere is:

-------------------
3

The Volume of a pyramid is:

(Area of Base) * (height)
------------------------
3

I think the reason your teacher wouldn't tell you where these formulas
come from is that you probably wouldn't understand. They are derived
from Calculus.  Here's the basic idea.

Let's do the volume of a sphere. It's a little more simple than a
pyramid, although the ideas are similar. I'm going to tell you how to
get the volume of a semi-sphere (or half of a sphere).  I think it's
obvious that once you get half of it, you can double your answer and
get the volume of the entire sphere.

Let's work with the top half of the sphere (it would look like a
bird cage).  Note that at each point the radius is a different length.
At the top of the bird cage the radius is zero, and at the bottom it
is some pre-determined number.  Just note that it is not constant.
Now... in your mind, make hundreds of little horizontal slices that
cut across the sphere.  Horizontal is NOT up and down.  It is from
left to right (across the horizon). These slices are so close
together that we're going to make an assumption. Pick a slice in the
middle, any one - it doesn't matter. Now look at it. It's going to
look like a disk, right? Except that the bottom will be a little
bigger than the top. What we do here is assume that because we made
so many slices, the radius at the top and the bottom of this disk is
the same. This is key!

We can now use geometry to find the area of this disk because it is
essentially just a cylinder with small height.  The volume of a
cylinder is Pi * r^2 * h. What we have now is the volume of one tiny
little disk.

Now comes the fun part.  Do the above procedure for each remaining
disk. See how long that could take? Luckily, there is a much easier
way to add up the disks. It involves Riemann Sums and Integration,
which you will study in Calculus.

The procedure for the pyramid is similar: slice it up into small pieces.
The difference? The pieces are not disks. They are now essentially in
the shape of a TV dinner, a square base with small height. Find the
volume (area of base * height) and add them up.

I hope this clears up your question.

-Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry

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