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I'm building a greenhouse with a curved roof. For sake of calculating how much material to buy, I need to find out the value of h in the diagram below.

Thanks in advance.

Roof of greenhouse

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    There is not enough information to answer your question. What kind of curve is it? Is it part of a parabola? – 2017-02-11
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    If it is a circular arch, the information is adequate for a (numerical) solution – 2017-02-11

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If you are just interested in the value, just look at this Wikipedia page and look at the formula for arc length. $$s=\left(h+\frac{c^2}{4h}\right)\arcsin\left(\frac{c}{h+c^2/4h}\right),$$ where $s$ is arc length and $c$ is the lenght of the 'base'. Plugging in your measurements, we have: $$13=\left(h+\frac{25}{h}\right)\arcsin\left(\frac{10}{h+25/h}\right).$$

To simplify, we can write it with an extra variable: $$D=h+25/h$$ (incidentally, this is diameter of the circle from which the arc is "cut out"). Then we need to solve: $$\sin(13/D)=10/D.$$ Simple numerical tools, as e.g. WolframAlpha allow us to calculate that: $$D\approx 10.6427,\ h\approx 3.5.$$

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    Why are you assuming that the curve is part of a circle? – 2017-02-11
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    Because the drawing suggests so ;-) Moreover, if it is NOT a part of the circle, the problem is hardly tractable. We have no information, what function defines the curve. Part of a circle is at least a good approximation. – 2017-02-12