Moment of inertia of an annular section (tube)
A tube keeps nearly all the stiffness of a solid shaft at a fraction of the mass.
Formulas
About this calculator
In an annular section the material sits where it works hardest — far from the axis. A tube whose inner radius is half the outer one gives up about 6% of its moment of inertia while shedding a quarter of the area.
Frequently asked questions
Ix = Iy = π(R⁴ − r⁴)/4, with R the outer and r the inner radius. The polar moment is twice that.
The moment of inertia scales with the fourth power of radius while area scales with the second. Removing material near the axis costs little stiffness and saves a lot of mass.