**Moment of Inertia Calculator**

The moment of inertia calculator will determine the second moment of area (also known as the moment of inertia of plane area or the area moment of inertia) of common geometric figures.

Before calculating the second moment of area, we need to understand the concept. So what exactly is the moment of inertia of plane area?

It is an explanation of how the area is distributed about the x-axis or the y-axis. When completing the calculations, the units will be to the 4th power.

When determining the moment of inertia along an axis, we generally consider the “base” as the distance across the x-axis, and the “height” as the vertical distance, along the y-axis.

Calculating the second moment of area of geometric figures can be confusing and time consuming by hand, so let this calculator do all the work for you. But the formulas will be given so if you actually need to calculate by hand, the calculator will be a great resource to check your work.

For common shapes the equations for moment of area are as follows:

**Triangle**

\(lx = \frac{(\text{height})^{3}(\text{width})}{36}\)\(ly = \frac{(\text{height})(\text{width})^{3}}{36}\)

**Rectangle**

\(lx = \frac{(\text{height})^{3}(\text{width})}{12}\)
\(ly = \frac{(\text{height})(\text{width})^{3}}{12}\)

**Regular hexagon**

(since all sides and all interior angles are the same)

\(Ix = Iy = 5\times \frac{\sqrt{3}}{16(\text{length of side})^{4}}\)

**Circle**

\(Ix = Iy = (\text{radius})^{4}(\frac{\pi}{4})\)

**Semicircle**

\(Ix = (\text{radius})^{4}(\frac{\pi}{8}-\frac{8}{9\pi})\)
\(Iy = (\text{radius})^{4}(\frac{\pi}{8})\)

**Ellipse**

\(Ix = (\frac{\pi}{4})(a)b^{3}\) where a = radius x and b = radius y

\(Iy = (\frac{\pi}{4})(b)a^{3}\)

It’s very important to note that these formulas work only if the shape in question features both axes cross the centroid. The centroid is located 2/3 of the way from each polygon vertex to the midpoint of the opposite side. It also refers to the center of mass of a geometric object.

The above formulas are great if the origin and the centroid coincide, but in reality that won’t always be the case. To find the moment of area in this case, consider the following information:

l : moment of inertia about the axis parallel to x-axis

A: area of the shape

lx: moment of area about the x-axis

a: distance between the parallel axes.

Now the moment of area formula is simply

\(I = Ix + A(a^{2})\)You can practice finding the moment of area by manually working out the calculations and then check your answers with our handy calculator. Or if you want a nearly instant answer, go straight to our calculator. Give it a try and you’ll be pleasantly satisfied with the results!