I believe that this is the answer.  If we model the roofline as a curve and then treat the roof as a surface of revolution, I believe that it produces the answer needed.

   Using points where x = -4.25, 0, 4.25 we get the curve:

   y = -32x^2/61 + 1092x/1037 + 5

   y = (-544x^2 + 1092x + 5185) / 1037

   Sorry for the large constants, but it was specified an exact answer was needed.

   V = ∫2 pi x f(x) dx  evaluated from zero to r

   V = (pi / 1037) (-272r^4 + 728r^3 + 5185r^2)

   I spared you the simplification process, but I am sure that you all know how to do this.

   V/thickness = A ,  in our case V/r = A

   A = (pi / 1037) (-272r^3 + 728r^2 + 5185r)

   Since our radius is 4.25 our area is 1683pi / 122 (exact)

The "ring wall" plus hemisphere will get close, I think (considering relative sizes). Maybe Allan could tell us how critical it is to get an exact answer. I emailed him to let him know I started a thread on his problem.

True.

Perhaps it means that there is a circular "ring wall" under the dome that is (5-4.25)=0.75 m high? If so, then multiply the height (0.75) by the circumference of the circle (2 × π × r) to get the area of the wall.

Dear Allan,

I am not sure if this is the exact answer for you ... but perhaps you can clarify it a bit more after reading

You mention a radius, so let us assume the dome is half a sphere.

The area of a sphere is 4 × π × r², and you want just half of that.

So, the pure half dome will have Surface Area =  2 × π × r² =  2 × π × (4.25)² = 113.5 m²