# Mohr’s Circle Calculator

URL: https://speedycalculator.com/calculators/physics/mohr-circle/
Last updated: 2026-10-30

> The Mohr's circle calculator finds the centre, radius, principal stresses, maximum shear stress and principal angle of a plane stress state, and the stresses on a plane rotated by any angle.

## Inputs

- Normal stress σx (MPa, kPa, Pa, GPa, psi, ksi)
- Normal stress σy (MPa, kPa, Pa, GPa, psi, ksi)
- Shear stress τxy (MPa, kPa, Pa, GPa, psi, ksi)
- Rotation angle θ (°, rad, arcmin, arcsec, mrad)

## Results

- Centre of the circle
- Radius (maximum shear stress)
- Principal stress σ₁
- Principal stress σ₂
- Principal angle θp
- σx′ on the rotated plane
- σy′ on the rotated plane
- τx′y′ on the rotated plane

## Formula

```
C = (σx + σy)/2,  R = √(((σx − σy)/2)² + τxy²),  σ₁,₂ = C ± R
σx′ = C + (σx − σy)/2·cos 2θ + τxy·sin 2θ,  τx′y′ = −(σx − σy)/2·sin 2θ + τxy·cos 2θ
```

## Explanation

Mohr's circle is a graphical tool that shows how normal and shear stresses change as you rotate the plane you look at. Its centre and radius give the principal stresses and maximum shear directly.

### How to use the Mohr's Circle Calculator

- **Normal stress σx**
- **Normal stress σy**
- **Shear stress τxy**
- **Rotation angle θ**

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

### How it works

Enter the normal stresses σx, σy and the shear stress τxy (with signs; tension positive), and optionally a rotation angle.

### Inputs explained

FieldWhat to enterNormal stress σxUnits: MPa, kPa, Pa, GPa, psi, ksiNormal stress σyUnits: MPa, kPa, Pa, GPa, psi, ksiShear stress τxyUnits: MPa, kPa, Pa, GPa, psi, ksiRotation angle θUnits: °, rad, arcmin, arcsec, mrad

### What each result means

- **Centre of the circle** = ([Normal stress σx] + [Normal stress σy])÷2
- **Radius (maximum shear stress)** = sqrt((([Normal stress σx] - [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- **Principal stress σ₁** = ([Normal stress σx] + [Normal stress σy])÷2 + sqrt((([Normal stress σx] - [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- **Principal stress σ₂** = ([Normal stress σx] + [Normal stress σy])÷2 - sqrt((([Normal stress σx] - [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- **Principal angle θp** = deg(atan2(2×[Shear stress τxy], [Normal stress σx] - [Normal stress σy]))÷2
- **σx′ on the rotated plane** = ([Normal stress σx] + [Normal stress σy])÷2 + ([Normal stress σx] - [Normal stress σy])÷2×cos(2×rad([Rotation angle θ])) + [Shear stress τxy]×sin(2×rad([Rotation angle θ]))
- **σy′ on the rotated plane** = ([Normal stress σx] + [Normal stress σy])÷2 - ([Normal stress σx] - [Normal stress σy])÷2×cos(2×rad([Rotation angle θ])) - [Shear stress τxy]×sin(2×rad([Rotation angle θ]))
- **τx′y′ on the rotated plane** = -([Normal stress σx] - [Normal stress σy])÷2×sin(2×rad([Rotation angle θ])) + [Shear stress τxy]×cos(2×rad([Rotation angle θ]))

### Example values

How the main result changes when **Normal stress σx** changes and everything else stays at the example values above:

Normal stress σx (MPa)Radius (maximum shear stress) (MPa)4042.426460508058.309510067.08212076.157716094.8683

## Worked example

Inputs — Normal stress σx: 80 MPa; Normal stress σy: -20 MPa; Shear stress τxy: 30 MPa; Rotation angle θ: 30 °.

Results — Centre of the circle = 30 MPa; **Radius (maximum shear stress) = 58.30952 MPa**; Principal stress σ₁ = 88.30952 MPa; Principal stress σ₂ = -28.30952 MPa; Principal angle θp = 15.4819 °; σx′ on the rotated plane = 80.98076 MPa; σy′ on the rotated plane = -20.98076 MPa; τx′y′ on the rotated plane = -28.30127 MPa.

## Key terms

- **Principal stresses**: Normal stresses on planes with zero shear.

## FAQ

**What do points on the circle represent?**
The normal and shear stress on planes at different orientations.

**Why is the angle doubled?**
A rotation θ of the plane corresponds to 2θ on Mohr's circle.

**Where is the maximum shear stress?**
At the top and bottom of the circle, 45° from the principal planes.

**What sign convention is used?**
Tension positive; positive τxy acts in +y on the +x face.


## 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/)