---
title: "Radius of gyration of angle section"
subject: "Applied Mechanics"
chapter: 4
key: "semester_1/applied_mechanics:radius-gyration-angle-section"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/radius-gyration-angle-section
site: BCT Engineering
---

# Radius of gyration of angle section

Applied Mechanics · Chapter 4: Center of Gravity, Centroid and Moment of Inertia

- Asked in: 2069 Chaitra, 8 marks
- Video: [Moment of Inertia, SY Civil | MOS | Radius of Gyration and Angle Section](https://www.youtube.com/watch?v=gCZ19evKPA4) — Vikas Institute Educomp
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/radius-gyration-angle-section
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-4

## Question

Determine radius of gyration (rx) of the angle section shown in figure below about centroidal x-axis. [Figure: L-shaped angle section, overall height 9 cm and overall width 6 cm, both legs 3 cm thick.]

## Short answer

Divide into 3×9 cm and 3×3 cm rectangles. Centroid at (2.25, 3.75) cm. I_xx = 249.75 cm⁴, A = 36 cm²; radius of gyration r_x = 2.63 cm.

## Answer

**Reading of the figure:** an L (angle) section, overall height 9 cm and overall width 6 cm, both legs 3 cm thick. Origin at the outer bottom-left corner.

Divide into: vertical leg 3 × 9 cm and horizontal leg 3 × 3 cm (the part of the bottom leg beyond the vertical leg, 6 − 3 = 3 cm long).

#### Centroid
| Part | A (cm²) | x (cm) | y (cm) | A·x | A·y |
|---|---|---|---|---|---|
| Vertical leg 3 × 9 | 27.00 | 1.50 | 4.50 | 40.5 | 121.5 |
| Horizontal leg 3 × 3 | 9.00 | 4.50 | 1.50 | 40.5 | 13.5 |

ΣA = 36.00 cm²
x̄ = 2.250 cm, ȳ = 3.750 cm

#### Moment of inertia about the centroidal x axis
I_xx = Σ [ b h³/12 + A (y − ȳ)² ]
```text
Vertical leg  : 3×9³/12 + 27 × (4.5 − 3.75)² = 182.25 + 15.19 = 197.44 cm4
Horizontal leg: 3×3³/12 +  9 × (1.5 − 3.75)² =   6.75 + 45.56 =  52.31 cm4
I_xx = 249.75 cm4
```
#### Radius of gyration
r_x = √(I_xx / A) = √(249.75 / 36) = √6.9375 = **2.63 cm**

## Check yourself

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

1. How is the angle section split?
   A. One 6x9 rectangle minus nothing
   B. Two 3x9 legs
   C. Two 3x6 legs
   D. Vertical leg 3x9 and horizontal leg 3x3 (not double-counting the corner)
   Answer: D. Overlapping the corner would count it twice, so the bottom leg is only the 3x3 beyond the vertical leg. (Key point: Division)

2. ȳ of the section is:
   A. 3.75 cm
   B. 4.5 cm
   C. 2.25 cm
   D. 3 cm
   Answer: A. (27*4.5 + 9*1.5)/36 = 3.75 cm. (Key point: Centroid)

3. r_x is obtained from:
   A. I_xx / sqrt(A)
   B. sqrt(I_xx / A)
   C. sqrt(I_xx * A)
   D. I_xx / A
   Answer: B. r = sqrt(249.75/36) = 2.63 cm. (Key point: Radius of gyration)

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