Rotate triangle around a rectangle

There are rectangle PQRS and a right-angled triangle ABC whose angle B is right as shown in a figure.

Triangle ABC is rotated for the surroundings of rectangle PQRS as follows.

The point A is at point P and point B is on the side PS at first.

① Triangle ABC is clockwise rotated around point B until the point C and the point S overlaps.

② Next, triangle ABC is clockwise rotated around point C until the point A and the point R overlaps.

③ Next, triangle ABC is clockwise rotated around point A until the point B comes on the side QR.

In this case, find the surrounding length of the portion by which triangle ABC moved.

The answer should round off the 2nd decimal place and should be found to the 1st decimal place.

Answer