A cone is created by rotating an isosceles right triangle with leg length 2 about one of its legs. Its surface area is $\pi$ times what numb

Here is the triangle before it is rotated. We can imagine that it will be rotated about the gray line.

We can see that...

the radius of the cone's base   =   r   =   2

the height of the cone   =   h   =   2

the surface area of the cone   =   π r (r + \(\sqrt{h^2+r^2}\) )

the surface area of the cone   =   π (2) (2 + \(\sqrt{2^2+2^2}\))

the surface area of the cone   =   2π (2 + √8)

the surface area of the cone   =   2π (2 + 2√2)

the surface area of the cone   =   π (4 + 4√2)