The first step in calculating the moment of inertia in an irregular geometric shape is dividing the shape into two or more regular shapes. For example, if you have a four-sided figure that has two 90-degree angles paired with an obtuse and acute angle, you can divide this area into a rectangle and a triangle. Dividing the area will allow you to use the mass moment of inertia equation to calculate the actual value of this irregularly shaped geometric figure.
After dividing the area of your irregular geometric shape, you must plug your data into the mass moment of inertia equation. The equation reads as follows: Inertia = ∫r^2 dm. In this equation, ∫ stands for line integral, while dm represents a small slice of mass that is parallel to the axis of the figure. The constant r stands for the distance of dm from the axis of the figure. Since you have two figures, from dividing your geometric shape in two, this equation must be performed on both shapes and added together.
Before you can solve the mass moment of inertia equation, you must reconfigure DM in terms of DX, using x as a variable. To do this, simply break down DM into the product of volume and density. In your equation, both of these numbers should be readily available to you, making it easy to get a complete figure from this. Because your figure is broken down into two separate figures, you will have to perform this multiplication step twice.
Once you have successfully modified DM in terms of DX, simply plug these values into your two mass moment of inertia equations. These two figures should be added together to calculate the mass moment of inertia. While you can calculate the moment of inertia on irregular geometries by hand using these equations, you can also use computer programs such as ProE or AutoCAD to help you come up with the data at a faster pace.