Possibility of a Figure with infinite area or perimeter

Question

Is there any figure possible with infinite area but finite perimeter or finite area but infinite perimeter? If yes, then what's the proof of its existence.

Answer 1

Finite area and infinite perimeter is easy: it is clear that the region between the curve $$ y = \sin{\left( \frac{1}{x} \right)} $$ and $x=1$, $x=0$ and, say, $y=2$ is finite (being a subset of the rectangle with bottom $y=-1$ instead of the sine curve and so on), but you can show that the length of the sine curve is unbounded (basically, it oscillates between $y=\pm 1$ infinitely often, so must have infinite length.

You can also consider the area bounded by the $x$-axis, the line $y=1$, and the curve $y=1/x^2$ (obviously the $x$-axis is infinitely long). This has a higher-dimensional analogue: Gabriel's horn.

On the other hand, the isoperimetric inequality says that the area is bounded above by the square of the length, so you can't have infinite area and finite perimeter.

In higher dimensions, the same thing happens (note the isoperimetric inequality generalises, and I mentioned Gabriel's horn above.