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Same Surface Area and Volume

Date: 03/28/2001 at 11:21:17
From: Zansia Solis
Subject: Surface Area and Volume

How can I find two rectangular prisms with the same surface area but 
different volumes?

Also how can I find two cylinders with the same surface area and 
different volumes?

Date: 03/28/2001 at 12:48:17
From: Doctor Peterson
Subject: Re: Surface Area and Volume

Hi, Zansia.

I would do the same thing in both cases. First choose the dimensions 
for one of the two objects, and find its area and volume. Now change 
the base; in the first case, perhaps divide the length by 2. Take the 
height as an unknown, and write an equation that says the surface area 
of the new shape is the number you got for the first shape. Solve this 
for h, and you will have a shape with the same surface area; check its 
volume to make sure it is different.

Here's an example: for an l x w x h rectangular prism,

     S = 2(lw + lh + wh)
     V = lwh

If I choose l = 4 cm, w = 3 cm, and h = 2 cm, I get

     S = 2(4*4 + 4*2 + 3*2) = 2(16+8+6) = 60 cm^2
     V = 4*3*2 = 24 cm^3

Now for my second cube, I will change l to 2 cm, keep w the same, and 
leave h unknown. Now

     S = 2(2*3 + 2h + 3h)
     V = 2*3*h

You want to solve the equation

     2(2*3 + 2h + 3h) = 60

and then check that 2*3*h is not 24.

If you don't know the formulas you need, click on "Formulas" in our 

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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