need to prove formula for height of anti-prism

[thxeveryone]

[This thread has been split off from a similar thread from three years ago.]

The difference in volume between OP's shape and an extruded 10 sided regular polygon seems to be 10 tetrahedrons (one is shown in green below). OP, could you calculate the volume of one such tetrahedron?

View attachment 25360

help I need to prove a formula for the height of the antiprism aka edge BC and I really like this approach (I don't understand any of the other ones I found online that require trig or calc and I don't understand the wolfram page) given the antiprism edges (AC, AD, and CD) are all 1. Is the green figure actually a tetrahedron (meaning all faces are same. edges are same, etc.)??

 

[Dr.Peterson]

I need to prove a formula for the height of the antiprism aka edge BC and I really like this approach (I don't understand any of the other ones I found online that require trig or calc and I don't understand the wolfram page) given the antiprism edges (AC, AD, and CD) are all 1.

Since your figure is equilateral, this formula in the Wolfram page applies:

For an equilateral antiprism d=a, so solving for h gives​

​

$h=\sqrt{1-\frac{1}{4}\sec^2\left(\frac{\pi}{2n}\right)}a$.​

The part above that (equation 4) tells how to derive this; if you need help doing that, please show what you have tried.

Is the green figure actually a tetrahedron (meaning all faces are same. edges are same, etc.)??

It is a tetrahedron (meaning it has four triangular faces), but not an equilateral (regular) tetrahedron. Those are not the same thing.

 

[thxeveryone]

The part above that (equation 4) tells how to derive this; if you need help doing that, please show what you have tried.

Hi Dr. Peterson, thank you for replying so quickly! For my previous question, I found a paper that explained it to me pretty well: https://www.pulsus.com/scholarly-ar...sis-of-regular-pentagonal-right-antiprism.pdf.

Solving for height of the top pentagonal pyramid:

Which leads into this (uses Rajpoot's formula for R0):