A bicycle has wheels of different sizes. On the rim of each wheel is a bug. As the bike travels, the bug on the smaller wheel traces a large number of small arches, and the bug on the larger wheel traces a smaller number of large arches. On a given bicycle trip, which bug travels the greater distance through space?

Remarkably, they travel the same distance. The arches are cycloids, and one revolution of a wheel produces one cycloid arch of length 8r, where r is the wheel’s radius. With each revolution the wheel travels a distance 2πr along the road. This means that during a bicycle journey of distance D, the wheel makes D/2πr revolutions, and the attending bug’s total distance through space is (D/2πr)(8r) = 4D/π. The rs have dropped out, which means that the size of the wheels doesn’t matter — the length of each bug’s path through space is 4/π times the length of the bicycle journey.

Via Doug Rohrer’s 1993 book Thought Provokers.

Somewhat similar: How much paint does my carousel need?