Can you dissect an equilateral triangle into pieces that can be rearranged into a square? If you think you can, you may have mastered the Pythagorean theorem.
Can you figure out how to draw a straight line without a reference edge? If so, you might just be the pioneer of the first straight line linkage.
Is it possible to drill a square hole? Yes. But only if you know what a Reuleaux triangle is.
Think you can determine how round a circle really is? Nice try. Checking the roundness of a manufactured object is way trickier than you might imagine.
The above mathematical complexities are broken down into an easy-to-consume format in the book, How Round Is Your Circle?, by John Bryant and Chris Sangwin. The authors use elementary geometry and trigonometry to reveal how past engineers applied mathematics to solve their problems, as was the case when Harry James Watts created his square-hole drill, or when Charles Nicolas Peacucellier figured out how to draw an exact straight line using a seven-piece planar linkage.
Click through to Bryant and Sangwin's site for more demonstrations and explanations. Pay attention, and the (geometrical) world around you just might begin to make a little more sense...