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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my 2 cents

| Comment 3 of 4 |

(In reply to re: Almost solution by Brian Smith)

The triangle and dodecagon are inscribed in the same circle so we need only find the ratio of the areas of a decagon to a trianlgle both inscribed in a unit circle as the ratio is the same.

The triangle has area 6*sin(60)*cos(60). The dodecagon has area 24*sin(15)*cos(15) so the ratio of those areas is 4*sin(15)*cos(15)/(sin(60)*cos(60)) ,and would be the ratio of the volumes of the dodecahedral pyramid to the triangular pyramid.

Posted by Charlie on 2023-07-06 17:37:01

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