Pappus's theorem is as follows:
First theorem:
The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C.
Second Theorem:
The volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F. (Note that the centroid of F is usually different from the centroid of its boundary curve C.)
Is there an intuitive way to see this?
I'll address the volume and you can address the surface area similarly. So let's consider the volume of a body of rotation (about the vertical axis, $z$). In cylindrical coordinates the volume is given by
$$V=\iiint r\ f(r,\theta,z)\ dr \ d\theta\ dz$$
Now, if $f=f(r,z)$ only, then
$$V=\int_0^{2\pi} \left[\iint r\ f(r,z)\ dr \ dz \right]\ d\theta=2\pi \iint r\ f(r,z)\ dr \ dz$$
Now, we know that the centroid is given by
$$C=\frac{\iint r\ f(r,z)\ dr \ dz}{\iint f(r,z)\ dr \ dz}$$
But the denomintor is just the area, $A$. Hence we end up with Pappus's $2^{nd}$ Centroid Theorem,
$$V=2\pi C A$$