---
title: "Define principal moment of inertia and principal axes"
subject: "Strength of Materials"
chapter: 2
key: "semester_3/strength_of_materials:principal-mi-and-axes-definition"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/principal-mi-and-axes-definition
site: BCT Engineering
---

# Define principal moment of inertia and principal axes

Strength of Materials · Chapter 2: Geometrical Properties of Sections

- Asked in: 2076 Chaitra, 2 marks; 2074 Chaitra, 4 marks; 2072 Chaitra, 4 marks; 2066 Bhadra (old course), 4 marks (asked 4 times)
- Video: [Principal moment of inertia and principal axis.](https://www.youtube.com/watch?v=1ZQzYlwm6tc) — Engineer Hyder's academy.
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/principal-mi-and-axes-definition
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-2

## Question

Define principal moment of inertias and principal axes.

## Short answer

Principal axes are a pair of mutually perpendicular axes through a point (usually the centroid) about which the product of inertia is zero, giving extreme moments of inertia. The principal moments of inertia are the maximum and minimum moments of inertia about these axes.

## Answer

#### Principal axes
For a plane area, the principal axes are the pair of mutually perpendicular axes through a point (normally the centroid) about which the **product of inertia is zero**. They are the axes about which the moments of inertia take extreme (maximum and minimum) values. For a section with an axis of symmetry, the axis of symmetry and the axis perpendicular to it are principal axes.

#### Principal moments of inertia
The moments of inertia about the principal axes are called the principal moments of inertia: the **maximum** $I_{max}$ about one principal axis and the **minimum** $I_{min}$ about the other. For a section with $I_x$, $I_y$ and $I_{xy}$ about the centroidal axes:

$$
I_{max,min} = \frac{I_x + I_y}{2} \pm \sqrt{\left(\frac{I_x - I_y}{2}\right)^2 + I_{xy}^2}
$$

The inclination of the principal axes to the x-axis is given by
$$
\tan 2\theta_p = \frac{-2I_{xy}}{I_x - I_y}
$$

Two directions, $\theta_p$ and $\theta_p + 90^\circ$, satisfy this equation. Note that $I_{max} + I_{min} = I_x + I_y$ (the polar moment of inertia is constant) and $I_{max}I_{min} = I_xI_y - I_{xy}^2$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What defines the principal axes of an area?
   A. The axes about which the area is zero
   B. A perpendicular pair through a point about which the product of inertia is zero
   C. Axes parallel to the edges of the section
   D. The axes through the corners
   Answer: B. Principal axes are perpendicular axes about which I_xy = 0; the moments of inertia about them are extreme values. (Key point: Definition of principal axes)

2. Which describes the principal moments of inertia?
   A. Both equal to the polar moment
   B. The maximum and minimum values of the moment of inertia, about the two principal axes
   C. The moments about the base of the section
   D. Two equal values for any section
   Answer: B. I_max is about one principal axis and I_min about the other, with I_max,min = (Ix+Iy)/2 +- sqrt(((Ix-Iy)/2)^2 + Ixy^2). (Key point: Principal moments of inertia)

3. For a section with one axis of symmetry, which pair of axes is certainly principal?
   A. The symmetry axis and the axis perpendicular to it
   B. Any two perpendicular axes
   C. Axes at 45° to the symmetry axis
   D. The edges only
   Answer: A. Product of inertia is zero when either axis is an axis of symmetry, so the symmetry axis and its perpendicular are principal. (Key point: Symmetry and principal axes)

4. Which relation holds between the principal moments and Ix, Iy?
   A. I_max - I_min = Ix + Iy
   B. I_max + I_min = Ix + Iy
   C. I_max x I_min = Ix + Iy
   D. I_max = Ix and I_min = Iy always
   Answer: B. The sum Ix + Iy (the polar moment) is invariant under rotation of axes, so I_max + I_min = Ix + Iy. (Key point: Invariants)

## Related in Strength of Materials

- [Principal MI about centroid and principal axes (2080 Baisakh)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-mi-2080bk) — Strength of Materials, chapter 2
- [Principal MI and axes of quarter-circle/triangle section (2078 Bhadra)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-mi-2078b) — Strength of Materials, chapter 2
- [Principal MI and axes, 120 cm section](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-mi-and-axes-orientation-120) — Strength of Materials, chapter 2
- [Principal axes of composite section (8, 12 cm)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-axes-orientation-composite-8-12) — Strength of Materials, chapter 2
- [Principal MI and axes of trapezoidal shaded area](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-mi-trapezoid-15-10-12) — Strength of Materials, chapter 2
- [Orientation of principal axes and MI of composite section](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-mi-two-triangles-composite-2066b) — Strength of Materials, chapter 2

## Related in other subjects

- [Moment of inertia by torsional pendulum](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amoment-of-inertia-torsional-pendulum) — Engineering Physics, chapter 1
- [Mass moment of inertia derivations](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Amass-moment-inertia-parallel-axis) — Engineering Mechanics, chapter 6
- [Composite body moment of inertia](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Arod-sphere-composite-inertia) — Engineering Mechanics, chapter 6
- [Inertia force and turning moment in an engine](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Areciprocating-engine-inertia-force-torque) — Theory of Mechanism and Machine I, chapter 8