# Coin Rotation Paradox Calculator

URL: https://speedycalculator.com/calculators/math/coin-rotation-paradox/
Last updated: 2026-10-30

> The coin rotation paradox calculator finds how many times a coin turns while rolling once around another coin (or inside a ring) without slipping.

## Inputs

- Diameter of the fixed coin (cm, m, mm, in, ft, yd, km, mi)
- Diameter of the rolling coin (cm, m, mm, in, ft, yd, km, mi)
- Rolling

## Results

- Number of rotations
- Distance travelled by the centre

## Formula

```
Rolling outside: N = 1 + D_fixed / D_rolling
Rolling inside: N = D_fixed / D_rolling − 1
```

## Explanation

Roll a coin around an identical coin and it turns twice, not once – the famous coin rotation paradox. The extra turn comes from travelling around the centre. A question on the 1982 SAT got this wrong, and none of the answer options was correct.

### How to use the Coin Rotation Paradox Calculator

- **Diameter of the fixed coin**
- **Diameter of the rolling coin**
- **Rolling**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the diameters of the fixed and rolling coins and choose whether the coin rolls around the outside or the inside. The answer counts full turns relative to a fixed observer.

### Inputs explained

FieldWhat to enterDiameter of the fixed coinUnits: cm, m, mm, in, ft, yd …Diameter of the rolling coinUnits: cm, m, mm, in, ft, yd …RollingChoose one: Around the outside, Around the inside

### Example values

How the main result changes when **Diameter of the fixed coin** changes and everything else stays at the example values above:

Diameter of the fixed coin (cm)Number of rotations1.21.51.81.752.4232.253.62.54.83

## Worked example

Inputs — Diameter of the fixed coin: 2.4 cm; Diameter of the rolling coin: 2.4 cm; Rolling: Around the outside.

Results — **Number of rotations = 2**; Distance travelled by the centre = 15.079645 cm.

## Key terms

- **Sidereal**: Measured relative to the distant stars.

## FAQ

**Why does the coin turn twice?**
Once from rolling along the circumference and once from going round the centre.

**What about a coin rolling inside a ring twice its size?**
It turns once.

**Is this related to the sidereal day?**
Yes – Earth spins 366.25 times per year relative to the stars but 365.25 relative to the Sun.

**Does it work for gears?**
Yes – planetary gears follow the same rule.


## Sources

- [Wolfram MathWorld – Coin paradox](https://mathworld.wolfram.com/CoinParadox.html)