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Volumes of a Cone and a Cylinder

```Date: 07/09/2003 at 07:19:00
From: Tony Francis
Subject: The volume of a cone and the volume of a cylinder

I am having trouble remembering formulae for areas/volumes. The volume
of a cylinder seems obvious, the area of the circle * length. The
volume of a cone appears to by one third of this. The fraction seems
to come up in other areas: e.g. a circle and a sphere (four thirds).

Can you tell me why the volume of a cone is a third of the volume of
a cylinder? I suspect it is something related to calculus but I don't
know where to start and the answer would probably help me remember the
formulae.
```

```
Date: 07/09/2003 at 08:06:40
From: Doctor Jaffee
Subject: Re: The volume of a cone and the volume of a cylinder

Hi Tony,

Draw a segment whose endpoints are (r,0) and (0,h), where r,h > 0.  If
you revolve this segment around the y-axis, it will form a cone whose
base is a circle of radius r and whose height is h.

To calculate the volume, sketch a representative disk somewhere
centered on the y-axis. The area of the surface of the disk shuld be
pi*x^2, where the x-number is the radius of the circle. The thickness
of the disk, then, is dy.

So, the volume of the cone will be the integral from 0 to h of
pi*x^2 dy. Of course, you'll have to express x in terms of y to allow
you to do the integration. The result should be V = pi*r^2*h/3.

Give it a try and if you want to check your solution with me or if you
have difficulties or other questions, write back to me and I'll try to

Good luck,

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

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