The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have:
$$
V_{C3}=\int_0^h \pi r^2 dz=\pi \int_0^h \frac{R^2}{h^2}z^2 dz=\pi \frac{R^2}{h^2} \frac{1}{3} h^3= \frac{1}{3}\pi R^2 h
$$
tha same result can be foud using the Pappus centroid theorem, rotating a right triangle with legs $h$ and $R$ around the cathetus $h$. Since the distance of the centroid from the axis of rotation is $R/3$, the theorem gives
$$V_{C3}=Ad=\frac{1}{2}Rh \cdot 2\pi\frac{R}{3}=\frac{1}{3}\pi R^2h$$
If we generalize such results to a hypercone in $\mathbb R^4$, the first procedure gives the result
$$
V_{C4}=\int_0^h\frac{4}{3}\pi r^3dz=\frac{4}{3}\int_0^h\left(\frac{R}{h}z \right)^3dz=\frac{1}{4}\cdot\frac{4}{3}\pi R^3h
$$
in accord to what we can find in Wikipedia (and in other internet resources). And this result can be generalized to $n-$dimension :
$$
V_{Cn}=\frac{1}{n}V_{S(n-1)}h
$$
($V_{S(n-1)}$ is the volume of the hypersphere of dimension $n-1$).
To generalize te second procedure I use the generalized Pappus Theorem as formulated in this page of Wikipedia:
Volume of $n$-solid of revolution of species $p$
= (Volume of generating $(n-p)$ solid)
$\times$ (Surface area of $p-$sphere traced by the
$p-$th centroid of the generating solid)
so, using the same triangle but rotating it on the surface of a $3-$sphere, I found:
$$ V_{C4}=Rh \cdot 4\pi\left( \frac{R}{3} \right)^2=\frac{4}{9}\pi R^3h$$
that is a different volume (and in $n-$dimension I have an analogous result).
Where is my reasoning wrong? I suspect that I have a misinterpretation of the generalized Pappus Theorem or that the rotation of a triangle around a $(n-1)-$sphere gives a different kind of cone with respect to the usual cone in n-dimension. If so, where I can find a classification of such kind of cones?
Best Answer
tl; dr: The formulas work out for a cone of height $h$ and base radius $R$ in four-space. The volume is indeed \begin{align*} \tfrac{1}{3}\pi R^{3}h &= (\tfrac{1}{2}Rh)(\tfrac{2}{3}\pi R^{2}) \\ &= (\text{area of generating triangle})(\text{area of sphere through the triangle's $p$-centroid}) \end{align*} for a suitable "$p$-centroid." This point is not the usual geometric centroid, however: Its location depends on $p$.
$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}\DeclareMathOperator{\Vol}{Vol}$Generalities: To start, here's a brief account of "volumes of rotation of species $p$." (This is doubtless the content of Sommerville's book, but Sommerville's notation is not how I visualize the geometry.) Let $p$ and $m$ be positive integers, and let $n = m + p + 1$. To describe what is meant by "revolving a region $D \subset \Reals^{m+1}$ about $p$-dimensional spheres," let $SO(p+1)$ denote the group of Euclidean rotations of $\Reals^{p+1}$, and decompose Cartesian $n$-space as \begin{align*} \Reals^{n} &= \Reals^{m} \times \Reals \times \Reals^{p} \\ &= \Reals^{m} \times \underbrace{\Reals^{p+1}}_{\text{$SO(p+1)$ acts}} \\ &= \underbrace{\Reals^{m+1}}_{\text{$D$ lives here}} \times \Reals^{p}. \end{align*} If $\Vec{x} \in \Reals^{m}$, $\Vec{y}_{0} \in \Reals^{p}$, $r$ is real, and $\Vec{y} = (r, \Vec{y}_{0}) \in \Reals^{p+1}$, the general element of $\Reals^{n}$ may be written $$ (\Vec{x}, \Vec{y}) = (\Vec{x}, r, \Vec{y}_{0}). $$ We're arranging that $D$ is contained in the half-space of $\Reals^{m+1}$ where $r \geq 0$ and $\Vec{y}_{0} = \Vec{0}$. A rotation $A \in SO(p+1)$ acts by $$ (\Vec{x}, \Vec{y}) \mapsto (\Vec{x}, A\Vec{y}); $$ the orbit of the point $(\Vec{x}, r, \Vec{0}) \in D$ is a $p$-sphere of radius $r$, specifically $\{\Vec{x}\} \times S^{p}(r) \subset \{\Vec{x}\} \times \Reals^{p+1}$.
Under this action, a volume element $d\Vec{x}\, dr$ of $D$ at $(\Vec{x}, r, \Vec{0})$ sweeps an infinitesimal volume $$ \Vol_{p} S^{p}(r)\, d\Vec{x} = (\Vol_{p} S^{p}) r^{p}\, d\Vec{x}. $$ The volume swept by revolving $D$ is the integral over $D$, $$ \Vol_{n}[SO(p+1)(D)] = \Vol_{p} S^{p} \int_{D} r^{p}\, d\Vec{x}. $$
This framework reduces to that of calculus if $p = m = 1$: In Cartesian coordinates $(x, r, y)$, the region $D$ lies in the half-plane $r \geq 0$ and $y = 0$; the group $SO(p + 1) = SO(2)$ revolves the $(r, y)$-plane about $\Reals^{m}$, a.k.a. the $x$-axis.
If we define $$ \bar{r}^{p} = \int_{D} r^{p}\, d\Vec{x}\bigg/\int_{D} d\Vec{x} = \frac{\Vol_{n}[SO(p+1)(D)]}{\Vol_{p} S^{p} \Vol_{m+1}(D)}, $$ then $\bar{r}$ is (by fiat, see Note at the bottom) the distance from the $p$-centroid to $\Reals^{m}$. Pappus' theorem is a formal triviality: \begin{align*} \Vol_{n}[SO(p+1)(D)] &= (\Vol_{p} S^{p}) \bar{r}^{p} \Vol_{m+1}(D) \\ &= \Vol_{p} S^{p}(\bar{r}) \Vol_{m+1}(D), \end{align*} the $p$-dimensional volume of the sphere of radius $\bar{r}$ times the $(m + 1)$-dimensional volume of $D$.
The four-dimensional cone: Here $m = 1$ and $p = 2$. The region $D$ may be taken to be the triangle $0 \leq x \leq h$ and $0 \leq r \leq Rx/h$. The $2$-centroid (really, its distance $\bar{r}$ from the axis) satisfies $$ \bar{r}^{2} = \frac{1}{\frac{1}{2}Rh} \int_{0}^{h} \int_{0}^{Rx/h} r^{2}\, dr\, dx = \frac{2R^{2}}{3h^{4}} \int_{0}^{h} x^{3}\, dx = \frac{R^{2}}{6}. $$ The sphere of radius $\bar{r}$ has area $4\pi \bar{r}^{2} = \frac{2}{3}\pi R^{2}$, so as claimed above, \begin{align*} (\text{area of generating triangle})(\text{area of sphere through $p$-centroid}) &= (\tfrac{1}{2}Rh)(\tfrac{2}{3}\pi R^{2}) \\ &= \tfrac{1}{3}\pi R^{3}h \\ &= \text{volume of the (hyper-)cone}. \end{align*}
Note: I don't see a natural mechanical interpretation of $\bar{r}$ Probabilistically, $\bar{r}^{p}$ is the mean of $r^{p}$, the $p$th power of the distance function from the "axis" (i.e., from $\Reals^{m}$). That is, $\bar{r}$ is the $p$-norm of the distance function $r$ over $D$, computed with respect to $(m + 1)$-dimensional Lebesgue measure.