Calculate the mass moment of inertia of rods, disks, cylinders, rings, spheres and plates, or the second moment of area of beam sections (rectangle, box, circle, tube, I-beam, triangle). Add an offset to apply the parallel-axis theorem and get section modulus and radius of gyration too.
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Distance from the centroidal axis to your axis (I = I_c + m d² or I_c + A d²)
Enter your values to see the result.
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How to use the Moment of Inertia Calculator
Choose the shape and axis: Mass shapes give I in kg·m², Area shapes give the second moment of area for beams.
Pick the length and mass units.
Enter the dimensions shown for that shape (and the mass for mass shapes).
Add a parallel-axis offset if your axis does not pass through the centroid.
Read the result and the extra properties, or copy them.
Mass moment of inertia formulas
Shape (axis)
Moment of inertia
Thin rod, length L (centre)
m L² / 12
Thin rod (one end)
m L² / 3
Solid disk or cylinder (central axis)
½ m R²
Solid cylinder (central diameter)
m (3R² + L²) / 12
Thin ring or hoop (central axis)
m R²
Thick hollow cylinder (central axis)
½ m (R² + r²)
Solid sphere (diameter)
⅖ m R²
Thin spherical shell (diameter)
⅔ m R²
Thick hollow sphere
⅖ m (R⁵ − r⁵) / (R³ − r³)
Rectangular plate b × h (perpendicular, centre)
m (b² + h²) / 12
Second moment of area (beam sections)
Section
Ix (centroidal)
Section modulus Sx
Rectangle b × h
b h³ / 12
b h² / 6
Hollow rectangle B × H, hole b × h
(B H³ − b h³) / 12
(B H³ − b h³) / (6H)
Solid circle, radius R
π R⁴ / 4 = π d⁴ / 64
π R³ / 4
Tube, radii R and r
π (R⁴ − r⁴) / 4
π (R⁴ − r⁴) / (4R)
I-beam d, bf, tf, tw
[bf d³ − (bf − tw)(d − 2tf)³] / 12
2 Ix / d
Triangle b × h
b h³ / 36
b h² / 24 (to apex)
The second moment of area grows with the cube of the depth, which is why beams are placed on edge and why I-beams put most of their material in the flanges, far from the neutral axis.
Frequently asked questions
What is the difference between mass and area moment of inertia?
Mass moment of inertia (kg·m²) measures resistance to angular acceleration and is used in rotation: τ = I α, E = ½ I ω². The second moment of area (m⁴ or in⁴) is a geometric property of a cross-section that measures resistance to bending, used in beam formulas like σ = M c / I.
What is the moment of inertia of a solid disk?
About its central axis, I = ½ m R². A 2 kg disk with a 0.25 m radius has I = 0.0625 kg·m². A thin ring of the same mass and radius has twice that, m R².
How do I calculate the moment of inertia of a rectangle section?
About the horizontal centroidal axis, I = b h³ / 12, where b is the width and h the height. A 2×4 board (1.5 × 3.5 in) on edge has I ≈ 5.36 in⁴; laid flat it is only 0.98 in⁴.
What is the parallel-axis theorem?
I = I_c + m d² for mass (or I_c + A d² for area), where I_c is about the parallel axis through the centroid and d the distance between the axes. A rod about its end: m L²/12 + m (L/2)² = m L²/3.
What is the section modulus?
S = I / c, where c is the distance from the neutral axis to the extreme fibre. The maximum bending stress is M / S, so a larger section modulus means a stronger beam in bending.
Are the I-beam results exact for rolled steel sections?
The calculator uses sharp-cornered rectangles for the flanges and web. Rolled sections have root fillets that add a few percent, so use the manufacturer's tables for final design values.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.