What is the formula for a solid sphere?

I = (2/5) × m × r², where m is the total mass in kg and r is the radius in meters. The axis passes through the center of the sphere.

What is the formula for a hollow spherical shell?

I = (2/3) × m × r², where m is the mass of the shell and r is its radius. This applies to a thin shell where all mass sits at radius r.

Why does the hollow shell have a larger coefficient than the solid sphere?

In a solid sphere, much of the mass is near the center where it contributes little to rotational inertia. The hollow shell concentrates all its mass at the outer radius, increasing the average distance squared and therefore the coefficient from 2/5 to 2/3.

What axis do these formulas apply to?

Both formulas assume rotation about any axis passing through the center of the sphere. Due to spherical symmetry, the moment of inertia is identical for every such axis.

What units does this calculator use?

Mass in kilograms (kg), radius in meters (m), and the result in kilogram-square-meters (kg·m²), the SI unit for moment of inertia.

Can I enter the radius in centimeters?

The calculator expects meters. Divide centimeters by 100 before entering (e.g., 15 cm = 0.15 m).

Moment of Inertia — Sphere

Calculate the moment of inertia of a solid sphere (I = 2/5 mr²) or a hollow spherical shell (I = 2/3 mr²) given mass and radius.

Sphere Parameters

Sphere Type

I = 2/5 × m × r²

Mass (kg)

Radius (m)

Result

Fill in the parameters and click Calculate.

Sphere Moment of Inertia Reference

Sphere Type	Formula	Notes
Solid Sphere	I = (2/5)mr²	Uniform density, axis through center
Hollow Spherical Shell	I = (2/3)mr²	Thin shell, all mass at radius r

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Summary

The moment of inertia of a sphere quantifies its resistance to rotational acceleration about an axis through its center. A solid uniform sphere uses I = (2/5)mr², while a thin hollow spherical shell uses I = (2/3)mr². Both formulas accept mass in kilograms and radius in meters, returning the result in kg·m².

How it works

Choose either Solid Sphere or Hollow Spherical Shell mode.
Enter the total mass of the sphere in kilograms.
Enter the radius in meters.
Click Calculate to apply the appropriate formula instantly.
Copy the result for use in further rotational dynamics calculations.

Use cases

Physics coursework on rotational dynamics and angular momentum.
Engineering design of spherical flywheels and gyroscopes.
Robotics and mechatronics — estimating joint and ball-joint inertia.
Comparing solid vs. hollow sphere configurations for a given mass.
Automotive and aerospace — ball bearing and spherical component analysis.
Sports science — analyzing spin and rotational energy of balls.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu