Rotating a quarter circle -- how long has a point traveled.

The difficult segment is the second where the quarter circle rolls $\frac\pi2$ arc on the ground. Assume unit radius, you may parametrize the path of $P$ with

$$x=t+\frac12\cos t ,\>\>\>\>\>y=-\frac12\sin t$$

Then, the path length of the second segment is

$$ \int_0^{\pi/2}\sqrt{(x_t’)^2 + (y_t’)^2 }dt=\int_0^{\pi/2} \sqrt{\frac54-\sin t}dt=1.19 $$

where the integral is elliptic, evaluated numerically.