Take a regular polygon and extend lines from each of its vertices at some angle so they all meet at the same point a little above the polygons centre to create a sort of pyramid. Is there a formula for finding the dihedral angle between the different planes of this pyramid? I would guess it would depend on base and height and number of vertices so let’s assume the base was an octagon of size 200mm and the height was 100mm. Is there a general formula for finding the dihedral angle with only this information?
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1Yes there is, I believe you can work it out using pythagoras and sines and cosines – student91 Jun 27 '23 at 12:52
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Any pointers regarding its method? – Steven Jun 27 '23 at 13:04
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I seem to remember a formula that gave the dihedral angle of two roof slopes say intersecting at ninety degrees to form a valley. 180 - inverse cosine ( cosine slope A * cosine slope B). I was hoping that there was a similar concise formula for this particular case. – Steven Jun 27 '23 at 13:44
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Then you are given some other information than you have given in the question. In that situation, you are given the (cosines of the) slope angles, in which case the formula will be more concise. The simplest formula there exists is (I believe) the spherical law of cosines, which says $\cos(\alpha)=\frac{\cos(a)-\cos(b)\cos(c)}{\sin(b)\sin(c)}$ when $a,b,c$ are the three angles that meet at a vertex, and $\alpha$ is the dihedral angle opposite angle $a$. But when you are given diameter and height of pyriamid, it takes work to get $a,b,c$. – student91 Jun 28 '23 at 09:26