



 
Huw, Your question is very much like the question that Daryl asked a while ago. I drew a diagram and labeled some points. The dimensions are in centimeters.
C is the centre of the circle that forms the arc and r is its radius. The triangle ABC is a right triangle, the length of AB is 24 cm, half the width of the arch, and the length of BC is r minus the the height of the arc above the top of the posts, which you want to be 12 cm. Thus, using Pythagoras' theorem
Expanding gives
and hence
Hence the length of BC is 30  12 = 18 cm and the angle BCA is cos^{1}(18/30) = 0.9273 radians. The length of an arc is r θ where θ is the angle that subtends the arc, measured in radians. Thus the length of half your arc is 30 0.9273 = 27.82 cm. Hence the arc at the top of your arch measures I hope this helps,  


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