Area Moment of Inertia Calculator

Reviewer: Dr. Sarah Jenkins, PE | Senior Structural Engineer | Last Updated: October 2023

Use this free Area Moment of Inertia Calculator to quickly determine the second moment of area (I) for common geometric shapes including rectangles, circles, and triangles. Essential for structural engineering and beam deflection calculations.

Select Shape

Width (b)

Height (h)

Inner Width (b_inner)

Inner Height (h_inner)

Outer Diameter (D)

Inner Diameter (d)

Please enter valid positive numbers.

Calculated Moment of Inertia (I_x)

0.00

Units⁴ (e.g., mm⁴, in⁴)

Step-by-Step Breakdown

Area Moment of Inertia Calculator Formula

The formula depends on the shape’s geometry relative to the neutral axis. Here are the most common formulas used by engineers:

Rectangle (Solid): I_x = (b * h³) / 12

Circle (Solid): I_x = (π * D⁴) / 64

Triangle: I_x = (b * h³) / 36

For a comprehensive list of geometric properties, refer to The Engineering Toolbox: Area Moment of Inertia.

Variables Explained

I_x (Area Moment of Inertia): A geometric property representing resistance to bending about the X-axis.
b (Width): The dimension parallel to the neutral axis.
h (Height): The dimension perpendicular to the neutral axis.
D (Diameter): The total width of a circular section.
d (Inner Diameter): The void width in a hollow pipe.

Related Calculators

What is Area Moment of Inertia?

The Area Moment of Inertia (also known as the Second Moment of Area) is a property of a 2D plane shape which characterizes its deflection under loading. It is typically denoted by the symbol I. In structural engineering, it quantifies how a beam’s cross-section resists bending.

A higher Moment of Inertia implies that the beam is stiffer and will deflect less under a given load. This is why I-beams are shaped the way they are—they maximize the material distance from the neutral axis, increasing I without adding excessive weight.

How to Calculate Area Moment of Inertia (Example)

Let’s calculate the moment of inertia for a standard rectangular wood beam that is 4 inches wide and 8 inches tall.

Identify the Shape: Solid Rectangle.
Identify Variables: Width (b) = 4, Height (h) = 8.
Select Formula: I = (b * h³) / 12.
Cube the Height: 8³ = 512.
Multiply by Width: 4 * 512 = 2048.
Divide by 12: 2048 / 12 = 170.67 in⁴.

Frequently Asked Questions (FAQ)

What units are used for Area Moment of Inertia?

The units are always length to the fourth power. Common examples include millimeters to the fourth (mm⁴), inches to the fourth (in⁴), or meters to the fourth (m⁴).

Why is the height cubed in the formula?

The height (dimension perpendicular to the axis of bending) has a much greater impact on stiffness than the width. Doubling the width doubles stiffness, but doubling the height increases stiffness by eight times.

What is the difference between I_x and I_y?

I_x represents resistance to bending around the horizontal X-axis (vertical loads), while I_y represents resistance to bending around the vertical Y-axis (lateral loads).

Does material strength affect Area Moment of Inertia?

No. Moment of Inertia is purely a geometric property. However, to calculate actual deflection, you must combine “I” with the Modulus of Elasticity (E), which is a material property.