Similar Pyramids and Measurement Ratios

Date: 08/10/2002 at 21:12:19
From: Deanna
Subject: Similar Pyramids

Dear Dr. Math,

The volumes of two similar pyramids are 27 and 64. If the smaller has 
lateral surface area of 18, how would I find the lateral surface area 
of the larger one?  

I have tried to go backward since V=1/3bh, so, 27/3 = 9, but I don't 
know what I'm really doing. I'm confusing myself the more I think 
about it. Can you please walk me through this?  

Thank you for taking time to read this.

Sincerely, Deanna

Date: 08/11/2002 at 02:42:11
From: Doctor Greenie
Subject: Re: Similar Pyramids

Hi, Deanna -

You COULD probably get to the answer the way you are going, but it 
would be a LOT of work, and it is work you don't need to do.

You can solve this problem using a very useful fact about similar 
figures that is stressed far too seldom in the math curriculum. The 
fact is this:

If the scale factor (ratio between linear measurements of two similar 
figures) is a:b, then

(1) the ratio between any area measurements of those two similar 
figures will be a^2:b^2

and

(2) the ratio between any volume measurements of those two similar 
figures will be a^3:b^3

In other words, if the ratio of any measurement in one dimension 
between two similar figures is a:b, then the ratio of any measurement 
in two dimensions between those two similar figures is a^2:b^2, and 
the ratio of any measurement in three dimensions between those two 
similar figures is a^3:b^3.

This principle works 'in any direction'. If you know that the ratio 
of some area measurement between two similar figures is a^2:b^2, then 
you know that the ratio between any two linear measurements will be 
a:b, and that the ratio between any two volume measurements will be 
a^3:b^3.

And - appropriate to your case - if you know that the ratio of some 
volume measurement between two similar figures is a^3:b^3, then you 
know that the ratio between any two linear measurements will be a:b, 
and that the ratio between any two area measurements will be a^2:b^2.

You are told that the volumes of the two similar pyramids are 27 and 
64.  The ratio of these two volume measurements is 27:64, which is 
3^3:4^3.  You therefore know that the ratio between any linear 
measurements on the two similar pyramids will be 3:4, and that the 
ratio between any area measurements on the two similar pyramids will 
be 3^2:4^2 = 9:16.

You are told that the lateral surface area of the smaller pyramid is 
18; to find the lateral surface area of the larger pyramid, we just 
need to solve a proportion using the ratio we now know between area 
measurements on the two pyramids.  So we have

     9   18
    -- = --
    16    x

and we find that the surface area of the larger pyramid is 32.

I hope 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: 08/11/2002 at 22:46:23
From: Deanna
Subject: Similar Pyramids

Dr. Math,

So would the conclusion be that since the volumes of the triangles are 
27 and 64, the lateral surface areas are 18 and 32?

Thank you so much for helping me.  
Deanna

Date: 08/12/2002 at 01:21:49
From: Doctor Greenie
Subject: Re: Similar Pyramids

Hi, Deanna --

Yes, that's right.

Thanks for taking the time to send us your thanks.

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