The Cauchy-Crofton formula on a plane

Question

For the complete tutorial for the formula you can check Do Carmo's Differential Geometry of Curves & Surfaces page through 42-46 at section 1.7.
The formula $$iint n(p,theta)dpdtheta=2 Len(alpha)$$
For finite length regular plane curve $alpha$ and $n$ is count of intersection of lines with curve.
My question is:
I understand that to find global property we have to define some measure on lines in the real plane. What I don't understand is what does this integral actually mean?
Elaboration of my confusion:
In the proof for a curve with length
$l$

he proves as above(at least gives a sketch)

So the integral(so called measure) $iint dpdtheta$ is evaluated according to some generic line.
But if I want to calculate for, lets say circle with radii $r$, I calculate it as if $p$ and $theta$ parametrize the circle. $2(Len(circ))=2(2pi)=iint limits_{0le ple r,0le thetale 2pi}underbrace{n}_{2}dpdtheta=2(2pi)$
So what is we are actually doing?

Ted Shifrin · Answer

You get $2(2pi r)$ in both cases, of course. You are integrating over the region in the space $(p,theta)$ of lines corresponding to lines that intersect the circle of radius $r$ centered at the origin. The lines with $p>r$ do not intersect the circle (so then $n=0$); the lines with $p=r$ intersect the circle once, tangentially (so then $n=1$); the lines with $p<r$ intersect the circle twice. So you get, indeed,
$int_0^{2pi}int_0^r 2,dp,dtheta = 2pi(2r) = 2(2pi r)$, which is twice the length of the curve.

The Cauchy-Crofton formula on a plane

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