Given a right triangle $ABC$ rotated $2\pi $, one full circle, around line $BC$. Find area of the formed cone.

Everyone knows that the formula for a cone includes a $\frac{1}{3}$ in it due to some integral calculus.

Why can the area of a cone not be visualized as a triangle rotated about an axis? Intuitively this makes sense. The area of a triangle is $\frac{1}{2}bh$. The distance it is rotated is $2\pi $. By that this should give us a formula $bh\pi $. This seems to work for a cone with $r=3$ and height $4$.

Everyone knows that the formula for a cone includes a $\frac{1}{3}$ in it due to some integral calculus.

Why can the area of a cone not be visualized as a triangle rotated about an axis? Intuitively this makes sense. The area of a triangle is $\frac{1}{2}bh$. The distance it is rotated is $2\pi $. By that this should give us a formula $bh\pi $. This seems to work for a cone with $r=3$ and height $4$.