This document provides formulas for calculating the moment of inertia for common shapes about different axes. It gives the moment of inertia formulas for rectangles, triangles, circles, semi-circles, and ellipses. The formulas treat the moment of inertia about axes through the centroid (overbar notation) and parallel axes theorems which allow calculating the moment of inertia about any other axis given the distance from the centroidal axis.

1. Useful Moment of Inertia Formulas
Note: In the table below, the overbar indicates the moment of inertia is taken about an axis that
passes through the centroid, denoted as ‘C’. Parallel axis theorems are:
2
AdII xx  2
AdII yy  yxAII xyxy 
Here, A is the area of the shape, d is the distance from the centroidal axis to the desired parallel
axis, and yx are the x and y distances of the centroid from the origin of the desired coordinate
frame.
Rectangle:
3
12
1 bhIx 
3
3
1bhIx

hbIy
3
12
1 hbIy
3
3
1
0yx
I Area = bh
Triangle:
3
36
1 bhIx

3
12
1 bhIx 
72
)2( 2
hsbb
Ixy

 bhArea
2
1

Circle:
4
4
1 rII yx 
0yx
I
2
rArea 
Semi-circle:
4
8
1 rII yx
 4
'
9
8
8
rIx 




 


0yxI
2
2
r
Area


Ellipse:
3
4
1 abIx  baIy
3
4
1
0yxI
abArea 
Cr
y
x
a
b
Cr
y
x
4r/3π
x`
Cr
y
x
b
h
x
x`C
h/3
s
y
x
y`
x`
b
h C
b/2
h/2