Applying the strategy of describing the surface area of a circle as a product of the ratio for the surface area of a triangle, reveals a consistency that also applies to the surface area of a cone. $A=1/2b * h$ or for a circle: $A=1/2c * r$ reduced to $A=1/2\pi (2r)*r=\pi r^2$.
SK 01:
By applying this perspective to the surface area of a sphere though, I end up with $\pi^2r^2$. This is resolved by the product of $4/\pi$ which is basically quantifying the bulge of the sphere. I wonder if there are similar clean ratios that resolve the properties of a 3-sphere? Measure the volume of an idea we cannot see or draw?
SK 02:
$A=c*1/4c$ or $A=2\pi r*1/2\pi r=\pi^2r^2$
Archimedes argued that the area of a sphere between two planes is simply proportional to their separation. This can easily be seen by examining the case where the planes are closely spaced. Then, by examining the case where two closely spaced planes include a great circle, the constant of proportionality must be equal to the circumference of the great circle. So take two parallel planes which just touch a sphere, thus spaced by the diameter, and the area of the sphere is equal to the diameter times the circumference of a great circle. In the same spirit as your first example, Archimedes argued that the volume of a sphere is equivalent to a cone whose base is the area of the sphere, and whose height is equal to the radius.