Moment of Inertia Calculator

Mass moment of inertia (kg m^2) measures how hard it is to rotate a physical body about an axis and depends on mass distribution. Area moment of inertia (m^4 or in^4), also called the second moment of area, depends only on the geometry of a cross-section and is used to predict how a beam resists bending. They share a symbol I, but the units and physical meaning are different. Use the type selector at the top to switch between them.

Because all of the hollow cylinder's mass sits at the outer rim, every bit of mass contributes the full r^2. In a solid cylinder some mass is close to the axis, where it barely contributes. The formulas show it directly: solid is (1/2) m r^2 while a thin hollow shell is m r^2, exactly twice as much.

The mass shapes assume the symmetry axis through the centroid unless noted. The thin rod has two variants: one with the axis through the center (perpendicular to the rod) and one through the end. The rectangular plate uses an axis perpendicular to the plate through its center. For area moments, the rectangle, circle, hollow circle, and triangle all use the centroidal axis parallel to the base.

It is the distance k at which all of the body's mass (or all of the cross-section's area) could be concentrated to give the same moment of inertia: k = sqrt(I / m) for mass cases, k = sqrt(I / A) for area cases. It is a single number that captures how spread out the mass or area is around the axis, useful for column buckling and rotational comparisons.

Because the cross-section would have zero or negative area, which is not a physical annulus. The outer diameter D must be strictly larger than the inner diameter d, otherwise the formula pi (D^4 - d^4) / 64 returns zero or a negative number.