Which mathematical object has a boundary of infinite length enclosing a finite area?

The Koch snowflake, built by repeatedly replacing the middle third of every edge with a triangular bump. Each step multiplies the perimeter by four thirds, so it grows without limit, while the added area shrinks fast enough to converge - the finished shape fits inside a circle you can draw.

That is not a paradox so much as a warning about the word length. The curve is continuous and has no tangent anywhere; you cannot draw a straight line touching it at a point, because every magnification reveals more corners. Mathematicians of the 1890s found such objects genuinely offensive - Hermite wrote of turning away in fear and horror from curves with no derivative.

Mandelbrot turned the problem into a measurement. Asked how long Britain's coastline is, the honest answer is that it depends on the ruler: measure in kilometres and you miss every inlet, measure in metres and the total grows, and it keeps growing as the ruler shrinks. The rate at which it grows is a fractional dimension, and it is a real property of the coast - which is why coastline lengths in different reference books disagree by large factors and none of them is wrong.