nice work, and yes, the 400 meters is as you wrote it, and yes, I thought that the Ferrari looks like a moving rectangle when seen from above, either with four-wheel steering or with a very slippery track and no steering needed at all.
I didn't use trigs.
Distance from the center of the semicircle to the middle of left side of the car (if driving counter-clockwise) = R.
Distance from the center of the semicircle to the right front / back corner = ((R+w)^2+(L/2))^(1/2)
So the width is of the track is ((R+w)^2+(L/2))^(1/2) - R.
which gives exactly the same result, of course.