I walk into a restaurant for lunch and my check comes to π dollars. The restaurant will only accept cash for the exact amount -- nothing more, nothing less. What are my options?

Find two gears, a big one and a small one. The small one's circumference should be equal to the big one's diameter. That means that when they are connected,  π rotations of the small gear equal one full rotation of the big gear. Put differently, one rotation of the big gear is equal to π times the circumference of the small gear.

Now, you tell the restaurant you'll work off your debt for a dollar an hour. You hook the two gears up to each other, and a clock's motor to the small gear. When the big gear has gone around one full time, the small gear will have gone around π times. So if you quit working as soon as that big gear finishes going around once, you'll have given them exactly π dollars worth of work!