---
title: "Radius of gyration"
subject: "Strength of Materials"
chapter: 2
key: "semester_3/strength_of_materials:radius-of-gyration"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/radius-of-gyration
site: BCT Engineering
---

# Radius of gyration

Strength of Materials · Chapter 2: Geometrical Properties of Sections

- Asked in: 2076 Asoj, 2 marks
- Video: [What is the Radius of Gyration?](https://www.youtube.com/watch?v=7kILaU2qOIU) — StructEd
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/radius-of-gyration
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-2

## Question

What is radius of gyration?

## Short answer

The radius of gyration k is the distance from an axis where the total area would be concentrated to give the same moment of inertia: k = √(I/A). It indicates the spread of area and is used in column design through the slenderness ratio L_e/k_min.

## Answer

The **radius of gyration** $k$ of an area about an axis is the distance from that axis at which the whole area could be assumed concentrated at a point so that the moment of inertia remains the same:

$$
I = Ak^2 \quad\Longrightarrow\quad k = \sqrt{\frac{I}{A}}
$$

Unit: length (mm, m). For example, $k_x = \sqrt{I_x/A}$ and $k_y = \sqrt{I_y/A}$; for a rectangle $b\times d$, $k_x = d/\sqrt{12}$ about the centroidal axis parallel to $b$. It measures how far the area is spread from the axis and is used in column design through the slenderness ratio $L_e/k_{min}$.

## Check yourself

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

1. What does the radius of gyration k of an area about an axis represent?
   A. The distance from the axis to the centroid of the area
   B. The distance from the axis at which the whole area could be concentrated at a point while giving the same moment of inertia
   C. The distance to the farthest point of the area from the axis
   D. The distance at which the area has zero moment of inertia
   Answer: B. k is defined by I = A k^2: the area lumped at distance k gives the same I. (Key point: Definition)

2. Which formula gives the radius of gyration?
   A. k = I/A
   B. k = sqrt(I*A)
   C. k = sqrt(I/A)
   D. k = I/A^2
   Answer: C. From I = A k^2 we get k = sqrt(I/A), and k has the unit of length. (Key point: Formula)

3. For a rectangle b × d, what is k about the centroidal axis parallel to b, and where is k used?
   A. k = d/12; it is used in beam shear stress calculation
   B. k = d/2; it is used for finding the centroid
   C. k = d/sqrt(3); it is used in torsion of shafts
   D. k = d/sqrt(12); it is used in column design through the slenderness ratio Le/k_min
   Answer: D. I = b d^3/12 and A = b d give k = d/sqrt(12); the slenderness ratio Le/k_min governs column buckling. (Key point: Use and meaning)

## Related in other subjects

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