Gauss's Law/Informal Explanation

Gauss's Law Informal Explanation
Consider a point charge $q$ at the center of a sphere of radius $r$.

From Surface Area of Sphere, the area of the sphere is $4 \pi r^2$.

From Coulomb's Law of Electrostatics, the electric field at the surface of the sphere is of magnitude $\dfrac q {4 \pi \varepsilon_0 r^2}$.

Hence the product of the area and the magnitude of the electric field is:
 * $4 \pi r^2 \times \dfrac q {4 \pi \varepsilon_0 r^2} = \dfrac q {\varepsilon_0}$

Hence the total magnitude of the electric field is independent of the radius.