Moment of Inertia (Circle) Calculator
The moment of inertia of a circular cross-section, properly its second moment of area, tells you how stiff a round bar, shaft or pipe will be when it bends. Like the rectangular case it is a purely geometric property that captures how the material is distributed about the bending axis, and it appears in every beam deflection and bending stress equation. For a solid circle the value is pi times the diameter raised to the fourth power, divided by sixty-four. Because the diameter is raised to the fourth power, a small increase in diameter produces a large jump in stiffness: increasing the diameter by half multiplies the moment of inertia by roughly five. This calculator takes the diameter of a solid circular section and returns the second moment of area in the fourth power of your length unit, for example mm to the fourth when you enter millimeters. The result is shown to two decimal places. Section property conventions and the related engineering standards are published by US federal agencies including the National Highway Traffic Safety Administration. Every figure is computed deterministically from the standard formula, shown in full below, with a worked example that reconciles exactly to the calculator so you can verify each step for yourself.
The second moment of area of a solid circle is pi d^4 / 64. For a 50 mm diameter bar, the moment of inertia is 306,796.16 mm4. Diameter dominates because it is raised to the fourth power.
Circular moment of inertia formula
I = pi d^4 / 64
I = second moment of area (length^4)
d = diameter of the solid circle
pi = 3.14159...
Raise the diameter to the fourth power, multiply by pi, then divide by sixty-four. The result is the second moment of area about any centroidal axis, which is the same in every direction for a circle. Because the diameter is raised to the fourth power, even modest increases in diameter sharply raise bending stiffness.
Worked example
A solid circular bar has a diameter of 50 mm and bends about an axis through its center.
- Raise the diameter to the fourth power: 50^4 = 6,250,000
- Multiply by pi: 3.14159265 x 6,250,000 = 19,634,954.08
- Divide by 64: 19,634,954.08 / 64 = 306,796.16
- The moment of inertia is 306,796.16 mm4
So the 50 mm bar has a second moment of area of 306,796.16 mm4. These are the calculator's default inputs, so the result above matches the widget exactly.
Moment of Inertia (Circle) Calculator: frequently asked questions
How do you calculate the moment of inertia of a circle?
For a solid circular cross-section, the second moment of area is I = pi d^4 / 64, where d is the diameter. A 50 mm bar gives pi x 50^4 / 64 = 306,796.16 mm4. The value is the same about any axis through the center because a circle is symmetric.
Why is diameter raised to the fourth power?
Bending stiffness depends on how far material lies from the neutral axis, and for a round section that distance scales with diameter. Integrating the contribution of every element across the area gives a fourth-power dependence on diameter, so a small diameter increase has a large effect.
How is this different from the polar moment of inertia?
The second moment of area pi d^4 / 64 governs bending. The polar moment of inertia, pi d^4 / 32, governs torsion (twisting) and is exactly twice this value for a solid circle.
How do I handle a hollow tube?
Subtract the inner circle from the outer: I = pi (D^4 - d^4) / 64, where D is the outer diameter and d the inner diameter. This page covers the solid circle; use that expression for a pipe or tube.
What is the circular moment of inertia formula?
For a solid circle, the second moment of area is I = pi d^4 / 64.
Official sources
- Structural section properties and engineering standards reference: US National Highway Traffic Safety Administration (NHTSA). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.