Parallel Axis Theorem
Steven Vukazich
San Jose State University
Recall the Definition of the Moment of Inertia
of an Area About an Axis
𝑑𝐴
y'
x'
𝐶
y
d
y = y' + d
x
𝐼$ = & 𝑦 ( 𝑑𝐴 = & 𝑦 * + 𝑑 ( 𝑑𝐴
Consider an axis x’ that
is parallel to the x axis
and passes through the
centroid of the area. The
distance between the
two parallel axes is d
Expand and Examine Terms
𝑑𝐴
y'
x'
𝐶
y
d
x
Moment of Inertia
of the area about
the x' axis
First moment of
the area about the
x' axis = 0
Area
𝐼$ = & 𝑦 * + 𝑑 ( 𝑑𝐴 = & 𝑦′( 𝑑𝐴 + 2𝑑 & 𝑦 * 𝑑𝐴 + 𝑑 ( & 𝑑𝐴
Parallel Axis Theorem
Centroidal Moment of Inertia
𝑑𝐴
y'
x'
𝐶
y
d
̅ + 𝑑(𝐴
𝐼$ = 𝐼$*
General Form
(
̅
𝐼 =𝐼+𝑑 𝐴
̅ = & 𝑦′( 𝑑𝐴
𝐼$*
Parallel Axis Theorem
x
If we know the moment of
inertia of a body about an
axis passing through its
centroid, we can calculate
the body’s moment of
inertia about any parallel
axis
Example Problem
y
ℎ
x
𝑏
Find the Moment
of Inertia of the of
the shaded area
about the x and y
axes shown. Use
the Parallel Axis
Theorem.
Note that this we have
already found Ix , Iy and the
location of the centroid for
this shape using
integration.
Moment of Inertia About Centroidal Axes
y
y'
ℎ
𝐶
2
𝑏
3
1
ℎ
3
b
Use Tabulated Solution for 𝐼 ̅
x
x'
̅ =
𝐼$*
1
𝑏ℎ5
36
̅ =
𝐼6*
1
ℎ𝑏 5
36
Moment of Inertia About Centroidal Axes
y
𝐼 = 𝐼 ̅ + 𝑑( 𝐴
y'
ℎ
𝐶
2
𝑏 = 𝑑6
3
1
𝐴 = 𝑏ℎ
2
1
ℎ = 𝑑$
3
b
1
1
(
5
̅ + 𝑑$ 𝐴 =
𝐼$ = 𝐼$*
𝑏ℎ + ℎ
36
3
(
1
𝑏ℎ
2
x
x'
1
̅ =
𝐼$*
𝑏ℎ5
36
̅ =
𝐼6*
1
ℎ𝑏 5
36
Moment of Inertia About the x Axis
1
1
(
5
̅
𝐼$ = 𝐼$* + 𝑑$ 𝐴 =
𝑏ℎ + ℎ
36
3
(
1
𝑏ℎ
2
y
1
1
5
𝐼$ =
𝑏ℎ +
𝑏ℎ5
36
18
3
1
5
𝐼$ =
𝑏ℎ =
𝑏ℎ5
36
12
Agrees with both the
tabulated solution and our
result from integration
ℎ
𝑏
x
Moment of Inertia About the y Axis
1
2
(
5
̅
𝐼6 = 𝐼6* + 𝑑6 𝐴 =
ℎ𝑏 + 𝑏
36
3
1
4
5
𝐼6 =
ℎ𝑏 +
ℎ𝑏 5
36
18
(
1
𝑏ℎ
2
y
ℎ
9
1 5
5
𝐼6 =
ℎ𝑏 = ℎ𝑏
36
4
Agrees with our result
from integration
𝑏
x