Quick q on gauss' law (cylinders)

[channel1]

Homework Statement

show that the E outside an infinitely long rod of radius R with a uniform charge density p is E = pR^2/2r(e_0)

Homework Equations

gauss' law EA=q/e_0

The Attempt at a Solution

I know how to solve this and get the correct answer but I don't totally understand it. Why is the height the same value on both sides of the gauss' law equation when the radii are different values?
ie, shouldn't there be a h and an H since there's a r and a R? obviously the answer is no since the heights cancel each other out, but why arent they different values?
why isn't the final answer E = pHR^2/2rh(e_0) instead of E = pR^2/2r(e_0) ?

 

[cupid.callin]

because the distance is measured from center not surface

 

[gomunkul51]

You are using Gauss Law, you are using it only because there is symmetry in this question.
All you measurements are symmetric. All radii start from the line of symmetry of the rod.
That way when you use Gauss' Law, the electric field E is the same at radius r from the center of symmetry.