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#1 2016-03-23 15:28:14

Monox D. I-Fly
Member
Registered: 2015-12-02
Posts: 857

[ASK] Distance from the Upper Base to the Center of Dodecahedron

I want to prove the formula of regular dodecahedron's volume. For this, I need to split the dodecahedron to 12 pentagonal pyramid with the same size. However, I don't know how to determine the height of those pentagonal pyramids. Any references I had encountered involving either golden ratio or incircle formula. Can we just determine their height using pure Pythagorean theorem? Trigonometry is also welcomed, but I still have some doubts about determining the angles where 3 faces of the dodecahedron meet because they provided no proof. Any help will be appreciated. Thanks.

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#2 2016-03-24 21:20:04

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,137

Re: [ASK] Distance from the Upper Base to the Center of Dodecahedron

o0PuD6G.gif

The red lines show one pyramid.  By drawing two green lines (vertex of a pentagon to the opposite midpoint) you can find the centre of a pentagon.  The blue line is then the height of the pyramid.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2016-03-25 00:47:11

Monox D. I-Fly
Member
Registered: 2015-12-02
Posts: 857

Re: [ASK] Distance from the Upper Base to the Center of Dodecahedron

Can we determine the the green and red line's heights without using inradius formula?

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#4 2016-03-25 21:27:14

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,137

Re: [ASK] Distance from the Upper Base to the Center of Dodecahedron

I can do the green using trig on a pentagon.  So far I haven't worked out the connection between the radius and the side length.  But I'm working on it smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2016-03-26 21:33:42

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,137

Re: [ASK] Distance from the Upper Base to the Center of Dodecahedron

Got a fresh idea.  The icosahedron and dodecahedron are duals.  So once you know about one the other should be relatively easy smile

I think the icosahedron is the easier one to tackle so here's a way to do it:

FsdOE.png

That rectangle E'C'CA is the thing to use after all!  You can calculate the length of E'C' by splitting the triangle D'E'C' to make  a 54-36-90 triangle and using trig.  E'A is the side length for the solid.  The diagonals E'C and C'A will cross at the centre of the solid so you can use Pythagoras to calculate E'C and hence calculate the perpendicular height of a pyramid.  The base area of a pyramid should be easy as they are all equilateral triangles.

Once you've got that sorted (let me know when you have) I'll explain how to transfer those measurements to a dual dodecahedron.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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