Hi Eric.

My first thought is to wonder if this is a spherical dome. If so, then these three numbers you provided should tell us. We can just consider it as an arc of a circle (two dimensional) to verify it.

The angle A formed by the chord and a line from the end of the chord to the top of the arc is, by trigonometry, is arctan(3/3.5). This angle is half the central angle. So the central angle of the arc is 2 arctan(3/3.5) = 81.2 degrees.

Now we know as well that the ratio 3/3.5 = 3.5/(r-3) due to similar triangles. This makes r = 3.5^2 / 3 + 3 = 7.0833 m.

If this is a circle, then the arc length of 10 m will be correct for a circle of radius 7.0833 m and a central angle of 81.2 degrees.

The arc length based on a circular model is 2(7.0833)(3.14159)(81.2/360) = 10.04 m.

That's quite close - so I can assume the dome is a spherical dome (technically we call it a spherical "cap").

Now on to the surface area.

The surface area is quite simple if you know the formula S = 2 pi R h, where R is the radius and h is the height, giving the surface area S. For the introductory calculus that gives us this formula, see http://mathworld.wolfram.com/Zone.html

Thus, for your situation,

S = 2 (3.14159) (7.0833) (3) = 133.5 square meters.

Note that this is one side of the dome. If you need to resurface both the inside and the outside of a dome shell, you would double this.

Cheers,
Stephen La Rocque