The point P is initially at the origin O. The circle, initially on the left, rotates right side. The point P follows the curved locus as shown. Reaches the top of the circle at P on the right. The diameter PP' gets inverted on the right side.
The distance rolled by the wheel is OP' on the x axis: π R , R = radius.
The diameter PP' = 2 R.
The triangle OPP' is a right angle triangle.
OP² = P'O² + P'P²
= (πR)² + (2R)²
= (π²+4) R²
The magnitude of displacement vector OP = √(4+π²) * R
The direction of OP = displacement vector = Tan⁻¹ PP' / OP'
= tan⁻¹ 2R/πR = tan⁻¹ 2/π = 32.48 deg.
OR, the slope of displacement vector = 2/π.