Math Help! Diameter of d-20, d-100 shaped room

[devilish]

Hi all,

My google-fu and geometry is in serious decline today.

I'd like to create rooms for a dungeon shaped like various
geometric figures - 20 sided, 30 sided, 100- or 120-sided.

First off, how do you know what shape each facet would be?
(for instance, why is a d20 facet a triangle while a d12 is a
pentagon?)

Second, if I know the shape of the facet and give it a number
(say a d20 facet is 10' on each side ) what would the diameter
of the shape be?

Please provide generic formulae (not just "here it is
for d20, and d100 ") as I need to create a whole bunch of rooms
of various shapes.

Many thanks in advance!
-D

PS For the inevitable "Why the heck does it matter? Just tell your players
it's 100 feet wide - they're not going to pull out calculators and check you!" :
I'm planning on submitting this for publication someday so *someone* will check my
math later. Thanks!

 

[Zelgar]

Here's a few formulae

Here's some formulae I found in my Engineering-in-Training Manual

Name, # of Faces, Form of Faces, Total Surface Area, Volume
tetrahedron, 4, equilaterial triangle, 1.7321 s^2, 0.1179 s^3
cube, 6, square, 6.0000 s^2, 1.0000 s^3
octahedron, 8, equilaterial triangle, 3.4641 s^2, 0.4714 s^3
dodecahedron, 12, regular pentagon, 20.6457 s^2, 7.6631 s^3
isoahedron, 20, equilateral triangle, 8.6603 s^2, 2.1817 s^3

The radius of a sphere inscribe within a regular polyhedron is

r = 3 x Volume of Polyhedron / Area of Polyhedron

Zelgar