Square Inscribed in a Circle Calculator
Calculate the largest square that fits inside a circle (inscribed square). Enter the circle's radius and get the square's side length, area, and perimeter — or enter the square's side to find the circle that just contains it. The key relationship is s = r√2.
Known measurement
Area of the largest square that fits inside the circle (= 2r²)
- 1
Side length
5 × √2 = 7.0711The inscribed square's diagonal equals the diameter, so side = radius × √2. - 2
Square area
7.0711² = 50
How does this calculator work?
The largest square inside a circle of radius r has side s = r√2 and area 2r². It covers exactly 2/π ≈ 63.66% of the circle. Enter the radius or the side length to get all related dimensions — the square's corners touch the circle, and its diagonal equals the diameter.
Formula
How this is calculated
A square is "inscribed" in a circle when all four of its corners (vertices) lie exactly on the circle. The diagonal of the square equals the diameter of the circle (2r), because the diagonal connects two diametrically opposite points on the circle.
Applying the Pythagorean theorem to the right triangle formed by two sides and the diagonal: d² = s² + s², so d = s√2 = 2r, giving s = r√2. The area of the inscribed square is s² = 2r². An inscribed square covers exactly 2/π ≈ 63.66% of the circle's area — a constant ratio, independent of size.
The problem also runs in reverse: given a square of side s, the circumscribed circle (smallest circle enclosing the square) has radius r = s√2 ÷ 2 = s/√2. This is used in design, manufacturing, and geometry problems where you need to fit or cut a square from a circular material.
Frequently asked questions
If the circle has radius r, the inscribed square has side s = r√2 and area 2r². Equivalently, the circle's diameter equals the square's diagonal: 2r = s√2.
Exactly 2/π ≈ 63.66%. The square area is 2r² and the circle area is πr², so the ratio is 2/π — a constant regardless of the radius.
An inscribed square has its corners on the circle (circle is outside). A circumscribed square has the circle touching its sides from inside, with radius r = s/2. The inscribed square is smaller: side r√2 vs diameter 2r for the circumscribed case.
Also known as
TG we-Calculate Editorial Team. (2026). Square Inscribed in a Circle Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/square-in-a-circle-calculator
TG we-Calculate Editorial Team. "Square Inscribed in a Circle Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/square-in-a-circle-calculator.
TG we-Calculate Editorial Team, "Square Inscribed in a Circle Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/square-in-a-circle-calculator
@misc{wecalculate_square_in_a_circle_calculator, title = {Square Inscribed in a Circle Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/square-in-a-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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