So inside every triangle there is a very special circle, sort of in prison there. It is called the inscribed circle. Study of this circle provides one of the beginnings to what is, in the business world, the task if packing things efficiently. A good question would be: for a circle of a given size, is there a triangle that can enclose it with less material (perimeter, or sum of the side lengths) than a square would, or could, use?
Dynamic geometry can open new worlds. For more, check out www.davemath.com
School students these days should be using technology, as should their teachers.
One such use of technology is as a motivator for the study of mathematics.
Here is a group of animated gifs created VERY EASILY in GeoGebra. They use very basic concepts: circle, rotation, segment, and midpoint. The graphs generated are generally not seen in schools until late in trig at the earliest, but they are accessible to very young students.
It is one way in which design can be an entry pint to the study of mathematics, rather than an exit point.
For more, see www.davemath.com