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### Volume of a Cone

```
Date: 5/9/96 at 19:4:26
From: Anonymous
Subject: Volume of a Cone

Hello Dr. Math,

I was wondering if you could help me with a geometry problem that I
have to do:

A coffee pot with a circular bottom tapers uniformly to a circular
top having radius half that of the base. A mark halfway up (by height)
says 2 cups. If the pot could be filled completely to the rim, how
much coffee would it hold?

Any information on how to solve this problem would be greatly
appreciated! Thanks.

```

```
Date: 12/11/96 at 01:02:47
From: Doctor Rob
Subject: Re: Volume of a Cone

Good problem!

Start by cutting a cross-section through the centers A and C of the
circular top and bottom of the pot. Then extend the side lines DF and
its counterpart until they meet at P. Connect the two centers A and C
with another line, and extend it until it meets the extensions of the
side lines at the same point P:

P
/|\
/ | \
/  |  \
/  A|   \
/----|----\D  Top
/     |     \
/      |      \
/      B|-------\E 2-Cup Line
/        |        \
/         |         \
-----------------------  Bottom
C          F

Then CF = 2*AD, and AB = BC, so DE = EF, using similar or congruent
triangles. Now you can show that PC = 2*AC = 4*BC using similar
triangles. You can compute the volume of the cones with vertex at P
and bases with centers at A, B, and C, using the formula for the
volume of a cone, V = (Pi*h*r^2)/3, where h is the height and r the
radius of the base. The volume of the large cone V(L) minus the volume
of the medium-sized cone V(M) will give the volume of 2 cups as marked
on the pot. The volume of the large cone V(L) minus the volume of the
small cone V(S) will be the volume you seek, call it x, measured in
cups. The equation is:

V(L) - V(S)   x
----------- = -
V(L) - V(M)   2

which you will then solve for x.  All the lengths such as BC and CF
from the diagram will cancel from the top and bottom of the fraction
on the left in this equation, so they need not be known.

I hope this helps.  If you still need more help, write back and we'll
give it another try.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

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

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