A Rectangular PrismDate: 11/26/95 at 14:31:1 From: Anonymous Subject: rectangular prism whose volume is greater than or equal to its surface area I have been doing a Geometry Scavenger hunt for school, and I have been pondering this for days, is it possible to have a rectangular prism that has a volume greater than it's surface area? I have tried to everything I can think of and I can't figure it out! If it is possible could you please give me the measurements! Date: 11/27/95 at 9:5:9 From: Doctor Ian Subject: Re: rectangular prism whose volume is greater than or equal to its surface area Hello! Strictly speaking, it doesn't make sense to say that a volume is greater than an area, any more than it makes sense to say that a weight is greater than a length. Volume and area are measures of different kinds of things, so they're not directly comparable. However, we can ask the question in this way: Is it possible to have a rectangular prism such that if V is the volume measured in cubic [whatever]s, and A is the surface area measured in square [whatever]s, then V > A? Asked this way, the answer is 'yes'. For example, suppose we have a cube that is 1 foot on each edge. The surface area is A = 6 * (1 ft * 1 ft) = 6 square feet while the volume is V = (1 ft * 1 ft * 1 ft) = 1 cubic foot Clearly, 6 > 1. On the other hand, if we measure lengths in inches rather than feet, the area is A = 6 * (12 in * 12 in) = 864 square inches while the volume is V = (12 in * 12 in * 12 in) = 1728 cubic inches Clearly, 864 < 1728. So which is 'larger', the surface area or the volume? We haven't changed the cube at all, just the units we use to measure length, which are arbitrary. If the relative size depends on which units we choose, can the question of which is larger really make sense? -Doctor Ian, The Geometry Forum |
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