Twist a shaft and every fibre in it resists — but not equally. The ones far from the centre line get strained the most and fight back the hardest, which means the section's resistance depends on how its metal is spread about the axis. That single geometric fact is bottled into one number, the polar moment of area.

For a solid round shaft it is $J = \dfrac{\pi d^{4}}{32}$ — read aloud J equals pi d to the fourth over thirty-two. $J$ is the polar moment of area, in millimetres to the fourth (mm⁴), and $d$ is the shaft's outside diameter in millimetres. Nothing else goes in: no load, no material, no length. J is pure shape. Solve it forwards from a diameter, or backwards from a listed J to the diameter that produced it.

The fourth power is the whole personality of this quantity. Grow a shaft from 50 mm to 60 mm — a 20% increase — and J rises by $1.2^{4}$, just over double. Slip and enter a radius where the diameter belongs and you are out by $2^{4} = 16$, in the direction that makes a healthy shaft look condemned. Read the drawing twice.

One more habit worth forming: J has a near-identical twin, $I = \dfrac{\pi d^{4}}{64}$, the diametral second moment of area, which is what bending uses. For a circle $J = 2I$ exactly, and the two sit one line apart in every section table. The 32 belongs to twisting; the 64 belongs to bending. Grabbing the wrong constant is the most common mistake in this entire chapter, and it is worth a factor of two every time.