Explanation of the Formula for the Volume of a Pyramid

Date: 06/16/2005 at 00:59:22
From: Adriano
Subject: Reason why the volume of a pyramid works that way

OK, I know that the volume of a pyramid is Base*Height/3, but can you 
please, please explain why?  I couldn't find the answer on _why_ the 
formula works in your archives.

I know that a pyramid is basically a prism with some of it "shaved" 
off, so I guess this is why you have to divide Base*Height by 3.

Date: 06/16/2005 at 09:57:06
From: Doctor Greenie
Subject: Re: Reason why the volume of a pyramid works that way

Hello, Adriano --

Here is a link to a page in the Dr. Math archives where this question 
is answered:

  Volume of a Pyramid
    http://mathforum.org/library/drmath/view/55041.html 

I found this page by searching the archives using the key words 
"volume pyramid".

On this page, there is a link to a site on the Internet containing a 
proof of this formula based on dividing a prism into 3 pyramids.  I 
myself find that figure hard to visualize, and the proof hard to 
follow.

Here is a different approach which uses a similar process and a 
somewhat informal approach to argue that the formula is base times 
height divided by 3.

Suppose we start with a cube, and from the center of the cube we 
draw lines to the 8 corners of the cube.  This divides the cube into 
6 congruent pyramids with square bases.

If the side of the cube is s, then the volume of the cube is s^3.

In each of the pyramids, the base is a square of side s, and the 
height is half of s.  If we were to multiply the area of the base 
times the height to get the volume of each pyramid, then we would find 
that the volume of each pyramid is

  (s^2)*(s/2) = (s^3)/2

But then the total volume of the 6 pyramids would be

  6 * (s^3)/2 = 3s^3

But we know the total volume of the pyramids is the volume of the 
cube, which is s^3.

Multiplying base times height to find the volume of a pyramid gives us 
a volume for the cube which is 3 times as large as it is supposed to 
be.  From this we can conclude that the volume of each pyramid is not 
base times height, but rather base times height divided by 3.

I hope all this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/17/2005 at 00:32:56
From: Adriano
Subject: Thank you (Reason why the volume of a pyramid works that way)

Thanks so much for answering my question!  Your explanation was way
better than the website you showed me.  I understand it better now.