# Slenderness Ratio Calculator

URL: https://speedycalculator.com/calculators/physics/slenderness-ratio/
Last updated: 2026-10-30

> The slenderness ratio calculator finds a column's radius of gyration and slenderness ratio K·L/r from its length, end conditions and cross-section shape, and classifies it as short, intermediate or long.

## Inputs

- Column length L (m, cm, mm, km, in, ft, yd, mi)
- End conditions
- Section
- Width b (smaller side / flange) (mm, cm, m, in, µm)
- Height h (mm, cm, m, in, µm)
- Wall / web thickness t (mm, cm, m, in, µm)
- Flange thickness t_f (mm, cm, m, in, µm)
- Outer diameter D (mm, cm, m, in, µm)
- Inner diameter d (mm, cm, m, in, µm)
- Minimum moment of inertia I (mm⁴, cm⁴, m⁴, in⁴)
- Area A (mm², cm², m², in²)

## Results

- Radius of gyration r
- Slenderness ratio K·L/r
- Column type (structural steel)

## Formula

```
λ = K·L / r,  r = √(I_min / A)
```

## Explanation

Slenderness decides whether a column fails by crushing or by buckling. Short stocky columns crush; long slender ones buckle at loads far below their crushing strength. Building codes limit the slenderness of steel and timber members.

### How to use the Slenderness Ratio Calculator

- **Column length L**
- **End conditions**
- **Section**
- **Width b (smaller side / flange)**
- **Height h**
- **Wall / web thickness t**
- **Flange thickness t_f**
- **Outer diameter D**
- **Inner diameter d**
- **Minimum moment of inertia I**
- **Area A**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the column length, choose the end conditions and section shape and enter the dimensions. The weaker axis is used.

### Inputs explained

FieldWhat to enterColumn length LUnits: m, cm, mm, km, in, ft …End conditionsChoose one: Fixed–fixed (theory 0.5), Fixed–pinned (theory 0.7), Pinned–pinned (theory 1.0), Fixed–guided (theory 1.0) …SectionChoose one: Rectangle, Hollow rectangle, Solid circle, Hollow circle …Width b (smaller side / flange)Units: mm, cm, m, in, µmHeight hUnits: mm, cm, m, in, µmWall / web thickness tUnits: mm, cm, m, in, µmFlange thickness t_fUnits: mm, cm, m, in, µmOuter diameter DUnits: mm, cm, m, in, µmInner diameter dUnits: mm, cm, m, in, µmMinimum moment of inertia IUnits: mm⁴, cm⁴, m⁴, in⁴Area AUnits: mm², cm², m², in²

### What each result means

- **Radius of gyration r** = [r]
- **Slenderness ratio K·L/r** = [End conditions]×[Column length L]÷[r]

### Example values

How the main result changes when **Column length L** changes and everything else stays at the example values above:

Column length L (m)Slenderness ratio K·L/r1.551.96152.2577.94233103.9233.75129.90384.5155.88466207.8461

## Worked example

Inputs — Column length L: 3 m; End conditions: Pinned–pinned (theory 1.0); Section: Rectangle; Width b (smaller side / flange): 100 mm; Height h: 150 mm.

Results — Radius of gyration r = 28.8675 mm; **Slenderness ratio K·L/r = 103.923**; Column type (structural steel) = Intermediate (inelastic buckling).

## Key terms

- **Radius of gyration**: √(I/A), how spread out the area is.

## FAQ

**What is a good slenderness ratio?**
Steel codes typically limit compression members to KL/r ≤ 200.

**Why use the minimum radius of gyration?**
The column buckles about its weakest axis.

**How do end conditions change it?**
Fixed ends reduce the effective length K·L.

**What are the class boundaries?**
They depend on the material – about 40 and 120 are common for structural steel.


## Sources

- [MIT OpenCourseWare – Mechanics of Materials](https://ocw.mit.edu/courses/2-001-mechanics-materials-i-fall-2006/)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)