perplexus.info :: Shapes : Dodecagon Based Pyramid

Dodecagon Based Pyramid (Posted on 2023-07-06)

Let PA₁A₂...A₁₂ be the regular pyramid, A₁A₂...A₁₂ is regular polygon, S is area of the triangle PA₁A₅ and angle between of the planes A₁A₂...A₁₂ and PA₁A₅ is equal to α.

Find the volume of the pyramid.

No Solution Yet Submitted by Danish Ahmed Khan

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Almost solution

| Comment 1 of 4

Call O the center of the dodecagon. The the area of the dodecagon is 3(OA1)^2 and the volume of the pyramid is V=(OA1)^2*OP.

Let M be the midpoint of segment A1A5. We then have sin(a)=OP/MP or OP=MPsin(a).

From triangle PA1A5 we have S=0.5*A1A5*MP or MP=2S/A1A5.

Sub this into the above to get OP=2S*sin(a)/A1A5.

Back to the base, A1A5A9 is an equilateral triangle so OA1=A1A5/sqrt3.

Finally, put this in the volume formula V=2(A1A5)S*sin(a)/3

The extra (A1A5) from squaring doesn't cancel, though so this solution is incomplete. I don't see how to get from this to a formula purely in terms of S and a.

Posted by Jer on 2023-07-06 13:27:03

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