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How Many Pyramids?

Date: 08/21/2003 at 21:51:56
From: John
Subject: Pyramids

If a pyramid is 2 feet tall, and crumbles into sand, how many 2-inch 
pyramids can be created from the sand used to create the original 

I figured it to be 650. (Based on height of 24", I assumed the 
base was 24" x 24", therefore there would be 12 x 12 = 144 for the 
first 2" row.  Next would be 22" or 11x11 = 121, etc.)

Date: 08/21/2003 at 22:41:18
From: Doctor Greenie
Subject: Re: Pyramids

Hi, John -

If you can make a picture in your mind of your solution, you will 
see that there is a lot of empty space between the layers of 
pyramids, so there will be many more than 650 pyramids.

This is a curious problem which is solved quite easily using an 
important geometric concept. That concept is the following:

Suppose two figures are scale models of each other (mathematically, 
we say they are similar). Suppose they have a ratio of similarity 
of a:b, meaning that the ratio of any measure of length in the two 
figures is a:b. Then

(1) the ratio of any measure of area between those two similar 
    figures is a^2:b^2; and
(2) the ratio of any measure of volume between those two similar 
    figures is a^3:b^3

In your problem, the original pyramid has a height of 2 feet, or 24 
inches; the small pyramids have a height of 2 inches. The ratio of 
those linear measurements is 24:2, or 12:1.

With the ratio of similarity being 12:1, the ratio of the volumes of 
the two pyramids is 12^3:1^3, or 1728:1.  That means the volume of 
the original pyramid is 1728 times the volume of each small pyramid; 
this in turn means you can make 1728 small pyramids from the sand of 
the original pyramid.

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

- Doctor Greenie, The Math Forum 
Associated Topics:
High School Polyhedra

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