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
| Field | What to enter |
|---|---|
| Situation | Choose one: Square inside a circle – from the radius, Square inside a circle – from the circle area, Circle inside a square – from the side |
| Circle radius r | Units: cm, m, mm, in, ft, yd … |
| Circle area | Units: cm², m², mm², in², ft², yd² … |
| Square side a | Units: 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) |
|---|---|
| 5 | 50 |
| 7.5 | 112.5 |
| 10 | 200 |
| 12.5 | 312.5 |
| 15 | 450 |
| 20 | 800 |
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)
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.
Frequently asked questions
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%).