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Area and Volume

The way they relate for different sized objects and some curious results of this fact:

It's a curious phenomenon but it's no mystery! And yet, it's something that isn't known by everyone. Scientists might assume everyone knows about it, but they don't. So, here it is explained: The interesting fact that scaling objects up or down gives different surface area to volume ratio. Yes, it's true: objects which are of different sizes have a different area to volume relationship even if they are the same shape!

Science fiction writers sometimes find this difficult to believe, or tricky to write into stories in which change of size is crucial to the plot. For example, it's a known fact that a person falling off a tall building will not be expected to survive their encounter with the ground. But if that person where by some science fiction gadget or gizmo shrunk down to one twentieth the size, then what? This takes some imagining, and if it was you in this sci-fi scenario, you might be even more worried at the prospect of going over the edge if you'd been made so small, because the building would appear to be twenty times higher. However, strange as it may seem, you'd almost certainly survive the drop from such a great height. (It has been mentioned anecdotally that rats have been known to survive jumping off high buildings).

How this works:

A cube 1ft across has a volume of 1 cubic ft, and has six sides each of which is 1 square ft, so that's a ratio of six to one. For every cubic foot there's six square foot of area.

Now take a cube 2ft across. This has a volume of 2 x 2 x 2 = 8 cubic ft, and each of the six sides has an area of 2 x 2 = 4 square ft, making a total of 24 sq ft. So, for every cubic foot there's three square foot of area.

It seems odd, as both cubes are the same shape! But they have a different amount of surface area for the volume they contain.

So, in general, something small has more area for its weight. The same shaped object scaled up has more weight per area. This phenomenon has many applications, some of which are mundane and some of which are more fanciful.

For example, cups of tea that are bigger than normal will not get cold so fast as normal sized cups of tea. Yet I've heard people say surprised things about even a pint pot of tea not having gone cold as soon as expected. (Picture shows an imperial gallon cup of tea). This kind of thing can be taken to extremes in the consideration of the geometrical considerations of the sun where although it's a flaming ball of gas in the sky it's not going to cool down for a very long time partly because it has much more volume per area in comparison to a hypothetically imagined ball of fire with a more domestic scale of area and volume. Another example is with Liquid Nitrogen in Thermos Flasks, the bigger the flasks, the better. This is just mentioned as a point of note on the page How to get Liquid Nitrogen.

The principle has implications in civil engineering, because structures such as bridges, ships, towers, cranes, and domes have to carry their own weight. A structure simply scaled up to twice as high/wide/etc has eight times the weight, but its supports are only four times as thick. Therefore the design of big projects requires some careful consideration to take account of this kind of effect. Generally, big structures are made using design which makes them very lightweight. For example, the Eiffel Tower weighs about nine thousand tonnes. Does that sound heavy for a tower which is a thousand feet high? To put that in perspective, if you made a perfect scale model of the Eiffel Tower 1 foot high it would weigh nine grammes! Sounds incredible, but it's simply that it's 1000 times scaled in height and 1000 times in width and 1000 times in depth, so is 1000 million times scaled in weight.

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