# Moments of Inertia

The moment of inertia, aka the angular mass or rotational inertia, of a rigid body determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation.

It is an extensive (additive) property: the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis).

vCalc's Moments of Inertia folder contains equations pertaining to the angular mass.

## Moments of Inertia Equations

• Area MOI - Triangle I
• Area Moment of Inertia - Triangle (Rtdot Angle Right)
• MoI
• MoI - Annulus around diameter
• MoI - Box Around Center
• MoI - Box around edge
• MoI - circular cone around axis
• MoI - circular cone around vertex
• MoI - circular plate (center)
• MoI - circular plate (diameter)
• MoI - cylinder - axis
• MoI - cylinder - diameter at end
• MoI - cylinder - perpendicular to axis
• MoI - cylinder about x-axis
• MoI - cylinder about y-axis
• MoI - ellipsoid around a tangent parallel to one axis
• MoI - ellipsoid around one axis
• MoI - hollow cylinder (axis)
• MoI - hollow cylinder (diameter at end)
• MoI - hollow cylinder (perpendicular to axis)
• MoI - hollow sphere (2 radii) around diameter
• MoI - hollow sphere (2 radii) around tangent
• MoI - solid sphere around diameter
• MoI - solid sphere around tangent
• MoI - spherical shell around diameter
• MoI - spherical shell around tangent
• MoI - thin rectangular plate through center
• MoI - thin rectangular plate through center parallel to one side
• MoI - Thin Rod Through Center
• MoI - Thin Rod Through End
• MoI - torus around axis in the plane
• MoI - torus around axis perpendicular to plane
• MoI - Box around center
• MoI - thin rod through center
• MoI - thin rod through end
• Moment of Inertia (No External Torque)
• Moment of Inertia - Cylindrical Shell
• Moment of Inertia - Ellipsoid (I_a)
• Moment of Inertia - Ellipsoid (I_b)
• Moment of Inertia - Ellipsoid (I_c)
• Moment of Inertia - Ellipsoid (axis_1 or axis_2)
• Moment of Inertia - Ellipsoid (axis_3)
• Moment of Inertia - Hollow Dodecahedron
• Moment of Inertia - Hollow Icosahedron
• Moment of Inertia - Hollow Octahedron
• Moment of Inertia - Hollow Sphere
• Moment of Inertia - Hollow Tetrahedron
• Moment of Inertia - Hoop (I_x, I_y)
• Moment of Inertia - Point Mass
• Moment of Inertia - Rectangular Plate (axis center)
• Moment of Inertia - Rectangular Plate (axis end)
• Moment of Inertia - Right Circular Cone (I_x, I_y)
• Moment of Inertia - Right Circular Cone (I_z)
• Moment of Inertia - Rod rotating about center
• Moment of Inertia - Rod rotating about end
• Moment of Inertia - Solid Cuboid (I_d)
• Moment of Inertia - Solid Cuboid (I_h)
• Moment of Inertia - Solid Cuboid (I_w)