Associated Topics || Dr. Math Home || Search Dr. Math

### Volume of a Prismatoid

```Date: 10/21/2008 at 10:05:49
From: Ravi
Subject: Volume of circular end pyramid

How can I calculate the volume of a pyramid for which the bottom end
is rectangular and the top end is circular?  I'm totally confused.

```

```
Date: 10/21/2008 at 10:49:23
From: Doctor Peterson
Subject: Re: Volume of circular end pyramid

Hi, Ravi.

This is not really a pyramid, but I picture it as a sheet-metal
transition from a round pipe to a rectangular duct, which is formed
by bending a series of triangles with one base in one end of the
figure, and the vertex in the other.  Is that right?

The volume can be calculated using the formula for a prismatoid,
which is the same kind of shape except that the top is a polygon
rather than a circle:

Prismatoid
http://mathworld.wolfram.com/Prismatoid.html

The formula is

V = h/6 (A1 + 4M + A2)

where A1 and A2 are the top and bottom areas, and M is the midsection
area.  In our case, if

r = radius of circle
L = length of rectangle
W = width of rectangle

it turns out that

A1 = pi r^2
A2 = LW
M  = (L/2 + r)(W/2 + r) - r^2 + pi r^2/4

and so

V = h/3 (pi r^2 + LW + Lr + Wr)

That is such a nice formula, I imagine I am not the first to write it!

See if this works for you.  If you have any further questions, feel
free to write back.

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/