Pappus centroid theorem and Hypercones.

centroidconvex-conehyperconesolid of revolutionvolume

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?

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.