Puzzles
This shape holds a finite amount of paint but infinite surface to paint
Take the curve y = 1/x for x ≥ 1 and spin it around the x-axis. The resulting trumpet-shaped solid has a finite volume — you could fill it with a finite amount of paint. But its surface area is infinite, so that same paint could never coat its entire inside. How can both be true at once?
Reveal the answer
Both facts are correct and don't actually contradict: filling the horn is a volume calculation (which converges to a finite number, π), while coating its surface is an area calculation (which diverges to infinity). The apparent paradox is about intuition, not the math — first described by Evangelista Torricelli in 1641, decades before calculus was formalized.
— Evangelista Torricelli, De solido hyperbolico acuto — 1641