Square in a Circle Calculator: Complete Guide to Inscribed Squares (2026)
Use our free square in a circle calculator to find the largest square inside any circle. Get the inscribed square side length, area, and diagonal with step-by-step solutions.
Try the free calculator
Put these formulas into practice with our instant, step-by-step Square in a Circle Calculator.
The square in a circle problem is a classic geometry challenge that asks: what is the largest square that can fit inside a circle? Whether you are cutting square parts from circular stock, designing architectural features, or solving a geometry homework problem, understanding the relationship between an inscribed square and its circumscribing circle gives you access to all related dimensions instantly. The formula s = r√2 connects the circle radius to the square side length in a simple, elegant relationship.
Key Takeaway
The largest square inscribed in a circle has side length s = r√2, area 2r², and diagonal equal to the circle diameter. The inscribed square occupies exactly 2/π ≈ 63.66% of the circle area.
What Is a Square Inscribed in a Circle?
A square inscribed in a circle is a square whose four vertices all lie on the circle. This is the largest square that can fit inside the circle — any larger square would have at least one vertex outside the circle. The inscribed square is unique up to rotation: you can rotate the square around the center, and it remains inscribed with the same dimensions.
The key geometric insight is that the diagonal of the inscribed square passes through the center of the circle and connects two diametrically opposite points. This means the diagonal of the square equals the diameter of the circle, which is the fundamental relationship that drives all the formulas.
The Formula: How to Calculate the Inscribed Square
The formula comes from the Pythagorean theorem. Since the diagonal of the square equals the diameter (2r), and for a square with side s the diagonal is s√2, we have s√2 = 2r, which gives s = 2r/√2 = r√2.
The square area is s² = (r√2)² = 2r². The circle area is πr², so the square occupies 2r²/(πr²) = 2/π ≈ 63.66% of the circle area. The remaining 36.34% is the gap between the square edges and the circle arc.
How to Use the Square in a Circle Calculator
- Enter the circle radius or diameter in the input field.
- Select whether you entered the radius or diameter.
- The calculator instantly shows the square side length, area, diagonal, circle area, gap area, and percentage of circle area used.
Worked Examples
| Circle Radius | Square Side | Square Area | Circle Area | % Used |
|---|---|---|---|---|
| 5 | 7.071 | 50 | 78.54 | 63.66% |
| 10 | 14.142 | 200 | 314.16 | 63.66% |
| 100 | 141.421 | 20,000 | 31,416 | 63.66% |
Notice that the percentage is always 63.66% regardless of the circle size. This is a universal constant of inscribed squares.
Common Mistakes to Avoid
- Confusing the square diagonal with the square side — the diagonal is s√2, not s.
- Forgetting that the diagonal equals the diameter — this is the key relationship.
- Using the wrong input — make sure to enter the circle radius, not the square side.
Frequently Asked Questions
- What percentage of a circle does an inscribed square use?
- Exactly 2/π ≈ 63.66%. This ratio is constant for all circle sizes.
- How do I find the square from the diameter?
- Use s = d/√2. The square side is the diameter divided by √2.
- Can I inscribe a rectangle other than a square?
- Yes, but the square has the largest area among all inscribed rectangles.
Conclusion
The inscribed square is a fundamental geometric shape with elegant properties. Use our free square in a circle calculator to find the largest square that fits inside any circle, along with all related dimensions and area calculations.