Moment of Inertia Calculator

Calculate second moment of area (I) for common structural shapes and composite sections.

Rectangular Section

I_x = bh³/12 | I_y = hb³/12

Width (b)

Height (h)

Unit

Hollow Section?

Results

Moment of Inertia (I_x)

—

Moment of Inertia (I_y)

—

Section Area —

Radius of Gyration (r_x) —

Radius of Gyration (r_y) —

Section Modulus (S_x) —

Section Modulus (S_y) —

Common Lumber Sizes:
2x4: 1.5" x 3.5" | 2x6: 1.5" x 5.5" | 2x8: 1.5" x 7.25"
2x10: 1.5" x 9.25" | 2x12: 1.5" x 11.25"

Circular Section

I = πd⁴/64 | I = πr⁴/4

Input Type

Diameter

Unit

Hollow Section (Pipe)?

Results

Moment of Inertia (I)

—

Section Area —

Radius of Gyration (r) —

Section Modulus (S) —

Polar Moment of Inertia (J) —

Common Pipe Sizes (Schedule 40):
3" pipe: OD 3.5", ID 3.068"
4" pipe: OD 4.5", ID 4.026"
6" pipe: OD 6.625", ID 6.065"

I-Beam / Wide Flange

I_x = (bd³ - (b-t_w)(d-2t_f)³) / 12

Total Depth (d)

Flange Width (b)

Flange Thickness (t_f)

Web Thickness (t_w)

Unit

Results

I_x (Strong Axis)

—

I_y (Weak Axis)

—

Section Area —

S_x (Strong Axis) —

S_y (Weak Axis) —

r_x —

r_y —

Common W-Shapes:
W8x31: d=8.0", b=8.0", t_f=0.435", t_w=0.285"
W10x49: d=10.0", b=10.0", t_f=0.56", t_w=0.34"

T-Beam Section

Total Height (h)

Flange Width (b_f)

Flange Thickness (t_f)

Web Width (b_w)

Unit

Results

I_x about Centroid

—

Section Area —

Centroid from Top (ȳ) —

S_top —

S_bot —

I_y —

Note: T-beams are common in reinforced concrete construction where the slab forms the flange.

Channel Section (C-Shape)

Depth (d)

Flange Width (b)

Flange Thickness (t_f)

Web Thickness (t_w)

Unit

Results

I_x (Strong Axis)

—

Section Area —

I_y (Weak Axis) —

Centroid from Web (x̄) —

S_x —

r_x —

Common Channels:
C8x11.5: d=8", b=2.26", t_f=0.39", t_w=0.22"
C10x15.3: d=10", b=2.60", t_f=0.436", t_w=0.24"

Composite Section (Parallel Axis)

Calculate I for built-up sections using the parallel axis theorem:
I_total = Σ(I₀ + A·d²)

Section 1 Area (A)

I₀ (about own centroid)

Distance from Reference (y)

Section 2 Area (A)

I₀ (about own centroid)

Distance from Reference (y)

Results

I_total about Centroid

—

Total Area —

Centroid (ȳ from ref) —

Σ(A·d²) —

Σ(I₀) —

Example: Two 2x10 boards stacked:
Each: A=13.875 in², I₀=98.93 in⁴
y₁=4.625", y₂=13.875"

Quick Reference - Common Section Properties

Section	Area	I_x	S_x	r_x
2x4 (1.5" x 3.5")	5.25 in²	5.36 in⁴	3.06 in³	1.01"
2x6 (1.5" x 5.5")	8.25 in²	20.80 in⁴	7.56 in³	1.59"
2x8 (1.5" x 7.25")	10.88 in²	47.63 in⁴	13.14 in³	2.09"
2x10 (1.5" x 9.25")	13.88 in²	98.93 in⁴	21.39 in³	2.67"
2x12 (1.5" x 11.25")	16.88 in²	177.98 in⁴	31.64 in³	3.25"
4x4 (3.5" x 3.5")	12.25 in²	12.51 in⁴	7.15 in³	1.01"
6x6 (5.5" x 5.5")	30.25 in²	76.26 in⁴	27.73 in³	1.59"