# Square in a Circle Calculator

URL: https://speedycalculator.com/calculators/math/square-in-a-circle/
Last updated: 2026-10-30

> The square in a circle calculator finds the largest square that fits inside a circle (from its radius or area), or the largest circle that fits inside a square, with the areas and the leftover space.

## Inputs

- Situation
- Circle radius r (cm, m, mm, in, ft, yd, km, mi)
- Circle area (cm², m², mm², in², ft², yd², km², ha, acre)
- Square side a (cm, m, mm, in, ft, yd, km, mi)

## Results

- Circle radius
- Square side
- Square area
- Circle area
- Leftover area
- Inner shape fills

## Formula

```
Square in circle: a = r√2,  square area = 2r²  (≈ 63.7% of the circle)
Circle in square: r = a/2,  circle area = πa²/4  (≈ 78.5% of the square)
```

## Explanation

How big a square beam can be cut from a round log? How large a round table fits in a square room? Inscribing one shape in the other answers both, and the leftover area tells you the waste.

### How to use the Square in a Circle Calculator

- **Situation**
- **Circle radius r**
- **Circle area**
- **Square side a**

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

### How it works

Choose the situation. For a square in a circle, the diagonal of the square equals the diameter of the circle; for a circle in a square, the diameter equals the side.

### Inputs explained

FieldWhat to enterSituationChoose one: Square inside a circle – from the radius, Square inside a circle – from the circle area, Circle inside a square – from the sideCircle radius rUnits: cm, m, mm, in, ft, yd …Circle areaUnits: cm², m², mm², in², ft², yd² …Square side aUnits: cm, m, mm, in, ft, yd …

### What each result means

- **Square area** = [Square side]^2
- **Circle area** = pi×[Circle radius]^2
- **Leftover area** = abs([Square area] - [Circle area])

### Example values

How the main result changes when **Circle radius r** changes and everything else stays at the example values above:

Circle radius r (cm)Square area (cm²)5507.5112.51020012.5312.51545020800

## Worked example

Inputs — Situation: Square inside a circle – from the radius; Circle radius r: 10 cm.

Results — Circle radius = 10 cm; Square side = 14.142136 cm; **Square area = 200 cm²**; Circle area = 314.159265 cm²; Leftover area = 114.159265 cm²; Inner shape fills = 63.662 %.

## Key terms

- **Inscribed**: Drawn inside another shape, touching it.

## FAQ

**What is the largest square in a circle?**
One whose diagonal is the diameter: side = r√2.

**How much of a log is wasted cutting a square beam?**
About 36.3%.

**What is the largest circle in a square?**
Its diameter equals the side.

**Which fills more: square in circle or circle in square?**
Circle in square (78.5% vs 63.7%).


## Sources

- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)