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

# Define product of inertia

Strength of Materials · Chapter 2: Geometrical Properties of Sections

- Asked in: 2075 Chaitra, 2 marks; 2075 Asoj, 2 marks (asked 2 times)
- Video: [Product of Inertia](https://www.youtube.com/watch?v=8_3sDlue0nI) — StaticsProf
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/product-of-inertia-definition
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-2

## Question

Define product of inertia.

## Short answer

Product of inertia I_xy = ∫ xy dA, units of length^4. It may be positive, negative, or zero (zero if one axis is an axis of symmetry). By the parallel-axis theorem, I_xy = Ī_xy + A x̄ ȳ.

## Answer

The **product of inertia** of an area about a pair of perpendicular axes $x$ and $y$ is the sum, over the whole area, of the products of each elemental area and its two coordinates:

$$
I_{xy} = \int xy\,dA
$$

Its unit is length$^4$ (for example mm$^4$). Unlike $I_x$ and $I_y$, it may be positive, negative or zero: it is zero if either axis is an axis of symmetry, and it is positive when most of the area lies in the first and third quadrants. By the parallel-axis theorem $I_{xy} = \bar I_{xy} + A\bar x\bar y$.

## Check yourself

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

1. The product of inertia of an area about axes x and y is defined as:
   A. The integral of y^2 dA
   B. The integral of xy dA
   C. The integral of (x + y) dA
   D. The integral of x^2 dA
   Answer: B. I_xy = integral of x y dA, the sum of each elemental area times its two coordinates. (Key point: Definition)

2. Which statement about the sign of I_xy is correct?
   A. It is always positive
   B. It is always negative
   C. It may be positive, negative or zero
   D. It is always zero
   Answer: C. Unlike Ix and Iy, which are squares, I_xy can take either sign, depending on where the area lies. (Key point: Sign)

3. When is the product of inertia zero?
   A. Whenever the area is a rectangle
   B. When either axis is an axis of symmetry
   C. Only for circles
   D. When the axes are parallel to each other
   Answer: B. If one axis is an axis of symmetry, positive and negative xy contributions cancel, so I_xy = 0. (Key point: Zero product of inertia)

4. The parallel-axis theorem for product of inertia is:
   A. I_xy = Ibar_xy + A(xbar + ybar)
   B. I_xy = Ibar_xy + A xbar ybar
   C. I_xy = Ibar_xy x A xbar ybar
   D. I_xy = Ibar_xy - A xbar^2
   Answer: B. The shift term is the area times the product of the centroid's offsets: I_xy = Ibar_xy + A xbar ybar. (Key point: Parallel-axis theorem)

## Related in Strength of Materials

- [Parallel axis theorem for product of inertia](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aparallel-axis-theorem-product-of-inertia) — Strength of Materials, chapter 2
- [First product of inertia of right-angled triangle](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aproduct-of-inertia-right-triangle) — Strength of Materials, chapter 2

## Related in other subjects

- [Stress in a section with product of inertia](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Aunsymmetrical-section-product-inertia) — Mechanics of Solids, chapter 7