I am wondering how to calculate surface of the church tower in the picture, for painting purposes. Especially, I am interested in the two 'onion-shaped' parts. I am thinking, that it is not really round, it consists of 6 equal parts, that are twisted, but would look like this on plane: like this:
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Thanks for any suggestions.
MATHEMATICAL ANSWER:
Referring to the picture: let $\theta$ be the angle $B\widehat{A}P$, in your case $\theta=\pi/6$.
$f(h)$ describes the shape of an edge of the dome as a function of the height, i.e. $f(h)$ is the distance of the edge from the central axis of the dome evaluated at a ceratin height $h$. Let's suppose that you know $f$ and let's suppose that the height of the dome is $H$. The length of the segment $AP$ is given by $f(h)\cos\theta$, so the length of the curve $OP$ is given by \begin{equation} s(h)=\int_0^h\sqrt{1+(f'(t)\cos\theta)^2}\ dt \end{equation} If we know how to express $BP$ in terms of $s$, by the means of a function $g(s)$, then the surface area of the portion of the dome is \begin{equation} A=2\int_0^{s(H)}g(s)\ ds \end{equation} We know that $BP=f(h)\sin\theta$, and that $ds/dh=\sqrt{1+(f'(h)\cos\theta)^2}$, so in the end I guess that \begin{equation} A=2\int_0^Hf(h)\sin\theta\sqrt{1+(f'(h)\cos\theta)^2}\ dh \end{equation}
PRACTICAL ANSWER:
as hinted in the comments, get a lot of paint (you will anyway use it for something else since you're a painter) and ask a for a lot of money (you can tell them you have to buy a lot of paint!)