Mass Moment of Inertia Calculator

By Dominik Czernia, PhD candidate

The mass moment of inertia calculator is a complex tool that helps estimate the moment of inertia of objects with different shapes. This physical quantity is otherwise known as the angular mass or rotational inertia. The moment of inertia is a characteristic property of a rigid body. It plays a similar role in the dynamics of rotational motion as the normal mass in the dynamics of translational motion. For example, the mass in the kinetic energy equation is replaced by the moment of inertia in the rotational energy equation.

You can use the mass moment of inertia calculator right now - just select a figure and enter its parameters. Or read on to learn what is the moment of inertia, what are its units, and how to calculate the moment of inertia. In the following text, we have also prepared a moment of inertia table with about 23 different figures. It contains almost all of the most common object shapes.

What is the moment of inertia? - moment of inertia units

Moment of inertia is the measure of the body's rotational inertia relative to a defined, fixed axis of rotation. It determines the torque which is needed for a desired angular acceleration. It is just like how mass determines the force needed for a desired acceleration. In other words, the moment of inertia tells us how difficult it is to put an object into rotation around a specific axis. Remember that the choice of axis is very important, the final moment of inertia value might strongly depend on it!

The physical dimension of the moment of inertia is mass * length². The SI unit of the moment of inertia is kilogram meter squared kg * m² and the imperial or US units is pound-foot second squared lb * ft * s² or pound foot squared lb * ft². With the mass moment of inertia calculator, you can perform calculations in any of those units you prefer.

Moment of inertia equation

The moment of inertia I of a material point is the product of its mass m and the square of the distance r from the axis of rotation. It can be expressed with the following moment of inertia equation:

I = m * r²

If you consider a body consisting of n material points, then the total moment of inertia is simply the sum of their moments of inertia:

I = Σ(mi * ri²)

where

  • Σ is the symbol of the summation. It sums all components from i = 1 to i = n,
  • mi is the mass of i-th material point,
  • ri is the distance of i-th material point from the axis of rotation.

However, for bodies with a constant distribution of mass, the summation in the above formula becomes an integral:

I = ∫(r² * dm)

where integration takes place over the entire volume V of the body.

Although integration is not always an easy task, there are many ready-made formulas for the moment of inertia of specific solids. You can select the figure from the list in this mass moment of inertia calculator or check the moment of inertia table in the next section.

The mass moment of inertia of a body that we just described, and the second moment of area are often confused. Remember that the mass moment of inertia units are kg * m² (lb * ft * s² or lb * ft²) and the second moment of area units are m⁴ (ft⁴).

Moment of inertia table

You have already learned what is the moment of inertia and how you can calculate it from its definition. In the table below, we have listed moment of inertia equations for simple objects with constant mass density, that can be selected in our mass moment of inertia calculator. When calculating moments of inertia, it is sometimes useful to exploit the parallel axis and perpendicular axis theorems to estimate moments of inertia about different axes.

Data and figures from Wikipedia

No. Description Figure and moments of inertia
#1 - ball Solid ball of radius r and mass m with axis of rotation going through its center.
The picture of a ball
Ball moment of inertia formula I = 2/5*m*r²
#2 - circular hoop Thin circular hoop of radius r and mass m with three axes of rotation going through its center: parallel to the x, y or z axes.
The picture of a circular hoop
Circular hoop moment of inertia formula Iz = m*r²
Circular hoop moment of inertia formula Ix = Iy = 1/2*m*r²
#3 - cuboid Solid cuboid of length l, width w, height h and mass m with four axes of rotation going through its center: parallel to the length l, width w, height h or to the longest diagonal d.
The picture of a solid cuboid
Solid cuboid moment of inertia formula (around four different axes)
#4 - cylinder Solid cylinder of radius r, height h and mass m with three axes of rotation going through its center: parallel to x, y and z axes.
The picture of a solid cylinder
Solid cylinder moment of inertia formula Iz = 1/2*m*r²
Solid cylinder moment of inertia formula Ix = Iy = 1/12*m*(3*r² + h²)
#5 - cylindrical tube Cylindrical tube of inner radius r₁, outer radius r₂, height h and mass m with three axes of rotation going through its center: parallel to x, y and z axes.
The picture of a cylindrical tube
Cylindrical tube moment of inertia formulae
#6 - cylindrical shell Cylindrical shell of radius r and mass m with axis of rotation going through its center, parallel to the height.
The picture of a cylindrical shell
Cylindrical shell moment of inertia fomula I ≈ m*r²
#7 - disk Thin solid disk of radius r and mass m with three axes of rotation going through its center: parallel to the x, y or z axes.
The picture of a thin solid disk
Thin solid disk moment of inertia formula Iz = 1/2*m*r²
Thin solid disk moment of inertia formula Ix = Iy = 1/4*m*r²
#8 - dodecahedron Solid and hollow, regular dodecahedron (twelve flat faces) of side s and mass m with axis of rotation going through its center and one of vertices.
The picture of a regular dodecahedron
Solid dodecahedron moment of inertia formula
Hollow dodecahedron moment of inertia formula
where
Golden ration formula - used to describe moments of inertia for solid and hollow dodecahedrons
#9 - ellipsoid Solid ellipsoid of semiaxes a, b, c and mass m with three axes of rotation going through its center: parallel to the a, b or c semiaxes.
The picture of a solid ellipsoid
Solid ellipsoid moment of inertia formula Ia = 1/5*m*(b² + c²)
Solid ellipsoid moment of inertia formula Ib = 1/5*m*(a² + c²)
Solid ellipsoid moment of inertia formula Ic = 1/5*m*(a² + b²)
#10 - icosahedron Solid and hollow, regular icosahedron (twenty flat faces) of side s and mass m with axis of rotation going through its center and one of vertices.
The picture of a regular icosahedron
Solid icosahedron moment of inertia formula
Hollow icosahedron moment of inertia formula
where
Golden ration formula - used to describe moments of inertia for solid and hollow icosahedrons
#11 - isosceles triangle An isosceles triangle of mass m, vertex angle 2β and common-side length L with axis of rotation through tip, perpendicular to plane.
The picture of an isosceles triangle
Isosceles triangle moment of inertia formula
#12 - octahedron Solid and hollow, regular octahedron (eight flat faces) of side s and mass m with axis of rotation going through its center and one of vertices.
The picture of an octahedron
Solid octahedron moment of inertia formula I = 1/10*m*s²
Hollow octahedron moment of inertia formula I = 1/6*m*s²
#13 - point mass Point mass m at a distance r from the axis of rotation.
The picture of a point mass
Point mass moment of inertia formula I = m*r²
#14 - rectangular plate Thin rectangular plate of length l, width w and mass m with axis of rotation going through its center, perpendicular to the plane.
The picture of a thin rectangular plate
Thin rectangular plate moment of inertia formula I = 1/12*m*(w² + l²)
#15 - regular polygon Plane regular polygon with n vertices, radius of the circumscribed circle R and mass m with axis of rotation passing through its center, perpendicular to the plane. Radius R can be expressed with side s.
The picture of regular polygons (n=3,4,5,6,7,8,10,12)
Regular polygons moment of inertia formulae
Formula for R that is used to describe regular polygons moment of inertia
#16 - right circular cone (hollow) Hollow right circular cone of radius r, height h and mass m with three axes of rotation passing trough its center: parallel to the x, y or z axes.
The picture of a hollow right circular cone
Hollow right circular cone moment of inertia formula Iz = 1/2*m*r²
Hollow right circular cone moment of inertia formula Ix = Iy = 1/4*m*(r² + h²)
#17 - right circular cone (solid) Solid right circular cone of radius r, height h and mass m with three axes of rotation passing trough its center: parallel to the x, y or z axes.
The picture of a solid right circular cone
Solid right circular cone moment of inertia formula Iz = 3/10*m*r²
Solid right circular cone moment of inertia formula Ix = Iy = 3/20*m*(r² + 4h²)
#18 - rod Rod of length L and mass m with two axes of rotation: about its center and one end.
The picture of a rod with rod moment of inertia formulae: Icenter = 1/12*m*L² and Iend = 1/3*m*L²
#19 - sphere Hollow sphere of radius r and mass m with axis of rotation going through its center.
The picture of a hollow sphere
Hollow sphere moment of inertia formula I = 2/3*m*r²
#20 - spherical shell Spherical shell of inner radius r₁, outer radius r₂ and mass m with axis of rotation going through its center.
The picture of a spherical shell
Spherical shell moment of inertia formula
#21 - tetrahedron Solid and hollow, regular tetrahedron (four flat faces) of side s and mass m with axis of rotation going through its center and one of vertices.
The picture of a tetrahedron
Solid tetrahedron moment of inertia formula I = 1/20*m*s²
Hollow tetrahedron moment of inertia formula I = 1/12*m*s²
#22 - torus Torus with minor radius a, major radius b and mass m with axes of rotating going through its center: perpendicular to the major diameter and parallel to the major diameter.
The picture of a torus
Torus moment of inertia formulae
#23 - two point masses Two point masses m₁ and m₂, with reduced mass μ, separated by a distance r with axis of rotation going through the center of mass and perpendicular to the line joining the two particles.
The picture of two point masses
The point masses moment of inertia formula I = μ*r²
Dominik Czernia, PhD candidate