Hong Life CalculatorsUnderstanding the square in a circle calculator — concept and principle
This calculator works out the square that fits exactly inside a circle, with all four corners on the circle. Enter the radius, the square's side, its diagonal or the circle's area, and it gives every other measure: side, diagonal, both areas, the leftover area, the share of the circle covered, the perimeter and the circumference. Use it for cutting a square beam from a log, design layouts, tiling and geometry homework.
How it is calculated
- The square's diagonal is the circle's diameter: d = 2r.
- Side s = r√2; conversely r = s ÷ √2.
- Square area = s² = 2r²; circle area = πr².
- Leftover area = (π − 2)r².
- Share covered = 2 ÷ π ≈ 63.66%.
- Square perimeter = 4s; circumference = 2πr.
Any one known quantity is first converted to the radius, then everything else follows.
The principle
Each corner of the inscribed square is on the circle, and the angle at each corner is 90°. A right angle inscribed in a circle always stands on a diameter (Thales' theorem), so the square's diagonal is a diameter. The diagonal of a square with side s is s√2 by the Pythagorean theorem, which links s and r. Because both areas are proportional to r², their ratio, 2r² ÷ πr² = 2/π, is the same for every size.
Worked example
Default input: radius r = 5.
- Diagonal = 2 × 5 = 10.
- Side s = 5√2 ≈ 7.07107.
- Square area = 2 × 25 = 50; circle area = 25π ≈ 78.5398.
- Leftover area = 78.5398 − 50 = 28.5398; share covered = 50 ÷ 78.5398 ≈ 63.66%.
- Perimeter = 4 × 7.07107 ≈ 28.2843; circumference = 10π ≈ 31.4159.
| Its value (length, or area in units²) | Square side s |
|---|---|
| 1.25 | 1.76777 |
| 2.5 | 3.53553 |
| 3.75 | 5.3033 |
| 5 | 7.07107 |
| 6.25 | 8.83883 |
| 7.5 | 10.6066 |
| 10 | 14.1421 |
How to read the result
The side is the largest square that fits in the circle. About 36% of the circle is left in the four curved segments, whatever the size. For cutting square timber from a round log, that leftover is the waste before sawing losses.
Common mistakes
- Setting the side equal to the diameter; it is the diagonal that equals the diameter.
- Using the diameter in place of the radius in s = r√2.
- Mixing up the inscribed square with the circumscribed square, whose side equals the diameter.
- Forgetting that the circle-area option expects square units.
Frequently asked questions
What is the largest square that fits in a circle of diameter 10?
Its side is 10 ÷ √2 ≈ 7.07, and its area is 50.
What percentage of a circle does an inscribed square cover?
2/π ≈ 63.66%, independent of size.
How is this different from a circle inside a square?
A circle inscribed in a square has diameter equal to the side and covers π/4 ≈ 78.54% of the square.
Sources: OpenStax, Prealgebra 2e, §9.3 (Pythagorean theorem) and §9.5 (circles)