# Moment of Inertia Calculator

Created by Luis Hoyos
Last updated: Aug 24, 2022

With this moment of inertia calculator, you can calculate the second moment of area of many common shapes.

In this calculator, "moment of inertia" refers to the moment of inertia of an area, also known as "second moment of area," "second area moment," "quadratic moment of area," or "area moment of inertia." Use our mass moment of inertia calculator if you're interested in the property that deals with the rotation of objects with mass.

In the rest of this article, we present the definition, importance, and formulas to calculate the moment of inertia of rectangles and other shapes.

## Moment of inertia definition

We can define the moment of inertia of a shape about an axis as the sum of each area differential multiplied by its squared perpendicular distance to the axis. For the arbitrary area below, the moments of inertia about the x and y axes are:

$I_x = \int_{A} y^2dA \\\ \\\ I_y = \int_{A} x^2dA$ An arbitrary shape. ρ is the radial distance to the element dA, with projections x and y on the axes. Attribution: [[:User:|IngenieroLoco]], CC BY-SA 4.0, via Wikimedia Commons

We can also define the second polar moment of area or polar moment of inertia ($J_0)$, which is similar but calculated about the "pole" or z-axis:

$J_0 = I_z = \int_{A} \rho^2dA$

We could also show that:

$J_0 = I_x + I_y$

## Why do we need to calculate the area moment of inertia?

The moment of inertia is a quantity that arises whenever a distributed load varies linearly and acts perpendicular to an area. When that occurs, the integrals $I_x = \int_{A} y^2dA$ or $I_y = \int_{A} x^2dA$ appear in the mathematical formulation of the problem. Examples are:

• To obtain the force caused by hydrostatic pressure - which varies linearly with depth - and its resultant moment, we need to calculate the area moment of inertia.
• In the mechanics of materials, calculating the second area moment is necessary to predict the strength and deflection of beams under bending. In our section modulus calculator, you can see how it's applied to the strength.
• Also, in the mechanics of materials, for beams under transverse shear, we need to calculate the area moment of inertia to obtain the maximum shear stress.
• With the polar moment of inertia, we can calculate the shear stress and strain caused by torsional loads in circular beams or shafts.

In most situations, we'll need the moment of inertia about the centroidal axis. That's why that's the axis about which this calculator finds the moment.

## Formulas to calculate the moment of inertia

As most of the shapes used in a structure are predefined, there are also predefined solutions to the moment of inertia to avoid embarrassing integrations.

In the following table, you can look at the formulas to calculate the moment of inertia of rectangles, triangles, circles, and many other common shapes about their centroidal axis.

Shape

Formula

Triangle Iₓ = bh³/36

Iᵧ = bh(b² - ab + a²)/36

Rectangle Iₓ = ba³/12

Iᵧ = ab³/12

Circle Iₓ = Iᵧ = (π/4)r

Semicircle Iₓ = (π/8 - 8/9π)r

Iᵧ = (π/8)r

Ellipse Iₓ =(π/4)ba³

Iᵧ = (π/4)ab³

Regular hexagon Iₓ = Iᵧ = 5√3a⁴/16 ≈ 0.541266a

Luis Hoyos
Shape
rectangle Width (b)
in
Height (a)
in
Ix
in⁴
Iy
in⁴
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