Parallel Axis Theorem for a Circle

Equations: for a circle about its centroidal axis, I_c = πd⁴/64. About any parallel axis, the parallel-axis theorem gives I = I_c + A·d′², where A = πd²/4 and d′ is the distance between the two axes. Values: d = 4 cm is the diameter. d′ = 0.5 cm is the offset from the centre to the new axis x′. Step 1: about the centre, I_c = π(4)⁴/64 = 256π/64 = 12.57 cm⁴. Step 2: area, A = π(4)²/4 = 12.57 cm². Step 3: transfer term, A·d′² = 12.57 × 0.5² = 3.14 cm⁴. Step 4: I = 12.57 + 3.14 = 15.7 cm⁴. 12.6 cm⁴ is the centroidal value only. 18.8 cm⁴ uses d′ instead of d′² (12.57 × 0.5 = 6.28). 25.1 cm⁴ is the polar moment πd⁴/32.