Consider the hex grid. Take a single hex on that grid; it has perimeter 0, and area 1. Then consider the ring of hexes around that hex; it has perimeter 6 and area 7. As you consider larger and larger hexagonal areas, an interesting pattern emerges:
size 1: perimeter=0, area=1 size 10: perimeter=54, area=271 size 100: perimeter=594, area=29701 size 1000: perimeter=5994, area=2997001 size 10000: perimeter=59994, area=299970001 size 100000: perimeter=599994, area=29999700001 size 1000000: perimeter=5999994, area=2999997000001 size 10000000: perimeter=59999994, area=299999970000001 size 100000000: perimeter=599999994, area=29999999700000001 size 1000000000: perimeter=5999999994, area=2999999997000000001
I've just learned that the complete sequence of hexagonal areas (not just the powers of ten) is registered in the OEIS as A003215 (aka the "crystal ball sequence for hexagonal lattice").
