Quite A Long Way Up

All we need to know is the area of the yellow region. Since every step has equal dimension, the length of the riser is equal to the length of the thread. We denote the length to be $x$ . If we see the figure, the area of the first rectangle (which is also a square) is $x^2$ . The next rectangle is $x(2x)=2x^2$ and so on until we reached $x(2016x)=2016x^2$ . We have now an arithmetic progression with $d=x^2$ , $a_1=x^2$ , $a_{2016}=2016x^2$ and $n=2016$ . So the area of the yellow region is

$s=\dfrac{n}{2}(a_1+a_{2016})=\dfrac{2016}{2}(x^2+2016x^2)=2033136x^2$

Now, $x=\dfrac{\sqrt{336}}{2016}$ . So $s=2033136\left(\dfrac{\sqrt{336}}{2016}\right)^2=168.08333$ .

Finally, the surface area of the stair-shaped prism is

$A=4(\sqrt{336})^2+2(168.08333)=$ $\boxed{1680}$