## Ice Cubed

There are few things more refreshing than a cold drink on a hot day.  But that coolness won't last forever; eventually thermodynamics rears its ugly head, warming our drinks and ruining our enjoyment.  It is an unfortunate fact that cold things in warm places don't stay warm for long.

To remedy this, everyone knows you can add ice to a drink to keep its temperature down.  This technique works because energy (in the form of heat) is transferred from your drink to the ice, raising its temperature and eventually converting it to liquid water.  This energy transfer can only happen where the ice makes physical contact with your drink, which is why the "crushed" option on ice dispensers is so great.  Slicing through a piece of ice increases its overall surface area without changing its total volume!

This may not seem obvious, but consider a few examples and you'll see that it's always the case.  For example, suppose you have a cube of ice that is one inch long on each side.  Since there are six square faces (top, bottom, left, right, front, and back), each with a surface area of one square inch, the cube has a total surface area of 6 square inches, and a volume of one cubic inch.

Now suppose you slice the cube in half as in the image above.  The amount of ice (i.e. the volume) doesn't change, but what about the surface area?  The six original sides of the cube will still be exposed, but now there are two new sides, represented in a darker shade of blue.  These new sides each have the same area as the left and right side of the original cube.  In other words, the total surface area of ice has increased from six square inches to eight - that's a 33% increase, from just one cut!

Surface area: increased.

This is a dangerous game, though, because the benefit of a cold drink is offset by the cost of a watery one.  Crush your ice too much and you may find that it dissolves too quickly, diluting your drink's flavor before you can finish it.  Even worse,  if you race against the melting ice and drink your beverage too quickly, you risk the onset of a dreaded brain freeze.

In other words, making the surface area of your ice as large as you can isn't necessarily the best approach.  Instead, it's probably best to strive for a balance between amount of coldness and level of dilution.  How that balance should be achieved, however, may depend in part on how long you'd like to enjoy your refreshment.

If recent trends are any indication, it would appear that avoiding a watery drink is more important than enjoying a chilled one.  This is evidenced by the proliferation of jumbo ice cubes in restaurants, bars, and the popularity of large ice cube trays online and in stores.  A single large ice cube in your drink will have a smaller surface area than many tiny ice cubes totaling the same volume.  So while the larger cube may cool your drink more slowly, it will water it down more slowly too, giving you the freedom to enjoy said drink at a more leisurely pace.

Of course, even among large cubes with a fixed volume, there's still a tremendous amount of freedom in what sort of shape is best for your ice.  For example, if all the shapes below contain the same volume of ice, can you determine which will have the smallest surface area?

Is there a shape that has a minimal surface area among all shapes of the same volume?  What about a shape with a maximal surface area?  And, on a more personal note, how much surface area do you think you'd prefer?

Teachers: we love thinking about ice so much that we wrote a lesson about it.  If you're a member, click here to access our latest lesson, Ice Cubed.