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 Question from Eric: I have recently been asked to resurface a dome sculpture for my local council but i'm having problems working out the area. Here are the dimensions. The height of the dome is 3m from the ground to the top of the arc. The arc itself from the ground rising up to the 3m point and back down, is 10m. The dome is 7m wide from one side to the other through the centre, at ground level. I hope there is enough detail here. It's been a long time since i was in a maths class. Thank you, hope to here from you soon!

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

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

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.