Volume of objects like hypercube / hypersphere : $V_{n}^{(m)}(r) = \dots$

Question

I am looking for some general form of equation for calculating volume for specific geometry objects.

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The main idea is to find :

$$ V_{n}^{(m)}(r) = \dots $$ Where:
$V$ - volume of object
$n$ - regular polygon
$m$ - dimension
$r$ - radius of described sphere

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It's easy to find equation for hypersphere, it is : $$ \lim_{n \to \infty} V_{n}^{(m)}(r) = \pi^{\frac{m}{2}} \frac{1}{\Gamma(\frac{m}{2} + 1)} * r^m $$ For $m=2$, general equation is : $$ V_{n}^{(2)}(r) = \frac{1}{2} n \sin(\frac{2 \pi}{n}) * r^2 $$

Thanks to Platonic solids there is equation for $m=3$ and $n=3$, $n=5$ : $$ V_{3}^{(3)}(r) = \frac{8 \sqrt{3}}{27} * r^3 $$ $$ V_{5}^{(3)}(r) = \frac{2 \sqrt{3} (5 + \sqrt{5})}{9} * r^3 $$

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In general it's easy to see that equation will have form : $$ V_{n}^{(m)}(r) = f(n, m) * r^m $$

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Is it possible to find exact equation ?

What worries me is limit in count of Platonic solids for $\text{3D}$ dimension case ($m=3$).
I wasn't also able to find any equation for other hyperobjects.

Answer 1

Tables I.i, I.ii, and I.iii, pp.292-295 of Regular Polytopes by H.S.M. Coxeter (3rd edition, Dover, ISBN-13 978-0-486-61480-9, ISBN-10 0-486-61480-8) have a column for the content (the name for $n$D equivalent of volume) of Platonic solids in 3D, their equivalents in 4D, and the 3 kinds in dimensions 5 and above. This is given in terms of the side length $2l$, another column gives the radii ${}_jR$ of the $j$D subpolytopes. There are 28 rows total, so forgive me for not typing them all up.

Table I.iii: The three regular polytopes in $n$ dimensions ($n \ge 5$)

Name	${}_j R/l$	$C/(2l)^n$
Regular Simplex	$\left(\frac{2}{j+1}-\frac{2}{n+1}\right)^{\frac{1}{2}}$	$\frac{1}{n!} \left(\frac{n+1}{2^n}\right)^{\frac{1}{2}}$
Cross Polytope	$\left(\frac{2}{j+1}\right)^{\frac{1}{2}}$	$\frac{2^{\frac{1}{2}n}}{n!}$
Measure Polytope	$\left(n-j\right)^{\frac{1}{2}}$	$1$