Question

A cone is generated when the region is bounded by the line x=y and the vertical line x=0 and x=r is rotated about the x-axis . Use the pappus theorem to show that , Surface area of the cone is given by S=√2πr²

Answer 1

Using the Pappus Theorem, we have that the surface area of the cone: $S=Ld$ , where $L$ the curve (blue segment) length and $d$ is the distance  traveled by the centroid (red point).

We can find $L$ using the Pythagoras' Theorem: $L=\sqrt{r^2+r^2}=\sqrt{2r^2}=r\sqrt{2}$

$d$ equals to the perimeter of the circle with the radius $r/2$ : $d=2\pi\frac{r}{2}=\pi r$ .

The we have $S=Ld =r\sqrt{2}\cdot \pi r=\sqrt{2}\pi r^2$