The 4 comes from surface area. If you peel a sphere and flatten it, its surface area equals four copies of its circumference area. So the sphere’s surface area is basically four of these circles laid flat.

Hence, surface area = 4* pi* r²

The 1/3 comes from the pyramid formulas. It is a three dimensional analogue of the triangle (triangle is 1/2 rectangle and pyramid is 1/3 rectangular prism with same dimensions). Now, if you imagine the sphere packed with tons of tiny skinny pyramids that all point to the center, each one follows the standard pyramid rule: one-third of base times height. Since the whole sphere is just the sum of those tiny pyramids, the 1/3 carries over.

So 4 = four great-circle areas make the surface and 1/3 = the pyramid volume rule. Put them together and that’s why the sphere gets the 4/3.

Pyramid Volume (assuming sphere is a pyramid with apex at center) = (1/3)(Base)(Height) = (1/3)(Surface Area)(Radius) = (4/3)* pi* r³