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)

]]>Maybe, this link would be of some help in determining the surface area. ]]>

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.

]]>And the highest point in the centre is 5 metres high.

The roof is an exact half of eclipse shape.

Contradiction!

]]>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

]]>Can I ask you an urgent question please.

I am trying to find out the external square metres of a dome shaped roof.

Its 8.5 metres accross ( 4.25mts in Radius.

And the highest point in the centre is 5 metres high.

The roof is an exact half of eclipse shape.

Help!

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