Solid circular sections appear as columns, piles, shafts and round bars. A circle has the same moment of inertia about every diameter, so it bends equally in all directions.
Formula
Worked example (millimetres)
Given
- Diameter D = 400 mm
Steps
- Area: A = π × D² / 4 = 125,663.7 mm².
- Moment of inertia: I = π × D⁴ / 64 = π × 400⁴ / 64 = 1,256,637,061 mm⁴ (125,663.7 cm⁴). It is the same about every diameter.
- Section modulus: Z = π × D³ / 32 = 6,283,185 mm³.
- Radius of gyration: r = D / 4 = 100 mm.
| Area, A | 125,663.7 mm² |
|---|---|
| Moment of inertia, Iₓ | 1,256,637,061 mm⁴ |
| Section modulus, Zₓ | 6,283,185 mm³ |
| Radius of gyration, rₓ | 100 mm |
Worked example (inches)
Given
- Diameter D = 8 in
Steps
- Area: A = π × D² / 4 = 50.27 in².
- Moment of inertia: I = π × D⁴ / 64 = π × 8⁴ / 64 = 201.1 in⁴. It is the same about every diameter.
- Section modulus: Z = π × D³ / 32 = 50.27 in³.
- Radius of gyration: r = D / 4 = 2 in.
| Area, A | 50.27 in² |
|---|---|
| Moment of inertia, Iₓ | 201.1 in⁴ |
| Section modulus, Zₓ | 50.27 in³ |
| Radius of gyration, rₓ | 2 in |
Try your own dimensions
The free calculator gives area, centroid, Iₓ, I_y, section modulus and radius of gyration in mm or inches.
Open the Moment of Inertia Calculator →Frequently asked questions
- What is the moment of inertia of a circle?
- I = π × D⁴ / 64, or π × r⁴ / 4 in terms of the radius. It is the same about any axis through the centre.
- What is the polar moment of inertia of a circle?
- J = π × D⁴ / 32, twice the area moment of inertia. It is used for torsion of shafts.
- What is the moment of inertia of a hollow circle or pipe?
- Subtract the bore: I = π × (D⁴ − d⁴) / 64, where D is the outside and d the inside diameter.