So the surface area is for the figure itself, and i just need to find the height. (geometry problem)

We need to cut it up and find the areas of each
ok so base=a^2
back sides are a^2 and a^2
total we have a^2+a^2+a^2=3a^2

we have now the 4 triangles, 2 of which are identical
the  identical triangles are the side ones which are base a and height 1/2a
area of triangle=1/2bh
so the areas of them are 1/4a^2, times 2 since 2 of those triangles, 1/2a^2

add
3a^2+1/2a^2=7/4a^2

now the very top triangle and the cut corner triangle

very top is legnth a and base a so 1/2 a^2 is area
add
3 and 1/2a^2+1/2a^2=4a^2

we also need the small rectangles on side which is 1/2a times a times 2 since 2 of them=a^2 so a^2+4a^2=5a^2
now to find the corner bit in terms of a
use some imagination and lines to find legnths, right triangles and hypotonuse
the very top side is a√2 (45,45,90 angle/triangle)
we need to find slant height which is (a√6)/2

now
area=1/2bh
area=((a^2)√3)

add that
5a^2+(a^2)√3=250.5
find a
undistribute the a^2
(a^2)(5+√3)=501
use divide both sides by (5+√3)
a^2=501/(5+√3)
take squaer root of both sides
a= aprox 8.6267094364168 in

I hope this is right (should be, but alot done in head so maybe incorrect or off)