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Average distance from circle's center to a point

> What is the average distance from a point inside a circle to the circle's center?

I came across this problem, and I've heard the solution is $\frac23R$ for the radius $R$.

So, I tried to tackle this myself:
* Let there be two concentric circles $C_1$ and $C_2$ with radii $R$ and $r$ ($R\gt r$).
* We pick a random point $P$ inside $C_1$. If there's a $50\%$ chance that $P$ is inside $C_2$, then $r$ is the average distance from $P$ to the center.

Then:
$$[C_1]=\frac12[C_2]$$
$$\not\pi r^2=\frac12\not\pi R^2\implies r=\frac12\sqrt2\cdot R$$

...which doesn't match the answer of $\frac23R$ (though, I should say $\frac12\sqrt2\approx0.707106781187$ which is quite close to $\frac23$).

Why is my reasoning wrong? If it's because $r$ isn't the average distance, why's that? I can't really make sense of this.

Thanks!