Need help ASAP with finding a formula for irregular "antiprism"

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thestruggle02

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Hi guys, I need help with finding a suitable formula that involves the VERTICAL HEIGHT (I mean the height from the centre of the pentagonal base straight up until it reaches the centre of the top pentagonal roof, just to be clear) of the antiprism as well as the length of the SIDES that make up the pentagonal base.

Is there any formula for this shape that I'm describing? I need help urgently as my assignment is due very soon.



1614250298668.png

someone suggested using this formula, BUT replacing the a^3 by a^2*h, where h is the vertical height. Does that work??



1614250281085.png
My antiprism it irregular so the sides at the bottom that make up the pentagonal base are NOT equal to the sides of the triangle faces that go around the structure

Please I really need help!!
 
thestruggle02 said:
Hi guys, I need help with finding a suitable formula that involves the VERTICAL HEIGHT (I mean the height from the centre of the pentagonal base straight up until it reaches the centre of the top pentagonal roof, just to be clear) of the antiprism as well as the length of the SIDES that make up the pentagonal base.

Is there any formula for this shape that I'm describing? I need help urgently as my assignment is due very soon.



View attachment 25353

someone suggested using this formula, BUT replacing the a^3 by a^2*h, where h is the vertical height. Does that work??



View attachment 25352
My antiprism it irregular so the sides at the bottom that make up the pentagonal base are NOT equal to the sides of the triangle faces that go around the structure

Please I really need help!!
Click to expand...
No, you can't just replace one of the a's with h. But you should be able to use that formula and multiply by the ratio of the height of your figure to the height of the regular antiprism. (I assume the only thing irregular about yours is that the lateral faces are isosceles rather than equilateral, and the bases are regular.)

So I would look for a relationship between h and a, by finding some right triangles.

I assume you got your information from https://en.wikipedia.org/wiki/Antiprism. There is more information (but only about the uniform antiprism, not the more general case you need) at https://mathworld.wolfram.com/Antiprism.html.

You might also be able to work out a formula yourself using calculus.
 
I just had a play with this. In summary, I highly recommend that you use @Dr.Peterson 's first suggestion...
Dr.Peterson said:
But you should be able to use that formula and multiply by the ratio of the height of your figure to the height of the regular antiprism...

So I would look for a relationship between h and a...
Click to expand...

...especially because there's a height formula on that Wolfram link (eqn 4). It assumes unit edge length, but you're safe to assume h will scale uniformly with a.
 
...further down the Wolfram page it almost gives the formula that OP is looking for. Eqn 30 on that page is incorrect since they missed an "n", it ought to be...

[math] V_{pyr} =\frac{1}{12}\color{red}n\color{black}a^2h\cot\left(\frac{\pi}{2n}\right) [/math]
if this is added to eqn 33 (in other words, Vpyr+Vbase) then the desired volume formula is obtained. Simplify by factoring. To check, I make the volume of h=2,a=2,n=5 approximately 14.846884858821
 
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