Square in a Circle Calculator

Find the side, diagonal, and area of the largest square that fits inside a circle — or the circle that circumscribes a given square — using d = D = s√2.

Quick Facts

Diagonal = diameter
d = D
An inscribed square's diagonal always equals the circle's diameter.
Square side formula
s = D/√2 = r√2
Follows from the Pythagorean theorem on two adjacent sides.
Square area formula
A = s² = 2r² = D²/2
The inscribed square covers about 63.7% of the circle's area.

Your Results

Calculated
Square side length
-
s = D/√2 = r√2
Square area
-
A = s²
Circle diameter (= square diagonal)
-
D = d = s√2
Circle radius
-
r = D/2

Ready

Choose what you know, enter the value and unit, then press Calculate.

Formula and Method for a Square Inscribed in a Circle

A square is "inscribed" in a circle when all four of its corners touch the circle's edge. Because a square's diagonal is a straight line from one corner through the center to the opposite corner, that diagonal is exactly a diameter of the circle: d = D. Every other measurement — the square's side, its area, and the circle's radius — follows from that single relationship, whether you start from the circle or from the square.

How the calculation works

Pick what you already know — circle radius, circle diameter, square side, or square diagonal — and this calculator converts it to a common diameter D first, then derives the rest. If you know the circle's radius r, the diameter is D = 2r. If you know the square's side s, the diagonal (and hence the diameter) is D = s√2, from the Pythagorean theorem applied to the right triangle formed by two adjacent sides (s² + s² = d²). Once D is known, the inscribed square's side is s = D/√2 = r√2, and its area is A = s² = 2r² = D²/2. The circle's radius is always half the diameter: r = D/2.

Common mistakes

  • Confusing side and diagonal: the square's diagonal (which equals the circle's diameter) is longer than its side by a factor of √2 ≈ 1.4142 — do not use the diagonal where the side is required, or vice versa.
  • Radius vs. diameter: the diagonal equals the full diameter, not the radius. Using r instead of D in the diagonal relationship halves the result.
  • Mixing units: keep the radius, diameter, or side length in one consistent unit before entering it — convert inches to feet, or centimeters to meters, first.

Real-world applications

  • Cutting the largest square panel, tile, or gasket from a circular sheet of material with minimal waste.
  • Sizing a circular frame, hoop, or lens mount that must fully enclose a square component.
  • Packaging and CNC/laser-cutting layouts that nest square parts inside round stock.
  • Geometry and trigonometry instruction, where this relationship is a standard worked example of the Pythagorean theorem.

Frequently Asked Questions

What is the relationship between a square inscribed in a circle and the circle's diameter?
When a square is inscribed in a circle, all four corners touch the circle and the square's diagonal passes through the center — so the diagonal of the square always equals the diameter of the circle: d = D.
How do you find the side length of the largest square that fits inside a circle?
Divide the circle's diameter by the square root of 2, or multiply the radius by the square root of 2: s = D/√2 = r√2. This comes from the Pythagorean theorem, since the diagonal (the diameter) is the hypotenuse of a right triangle formed by two adjacent sides.
How do you find the radius of the smallest circle that circumscribes a square?
Take half the square's diagonal: r = s√2/2 = s/√2, where s is the side length. The diagonal itself is s√2 and equals the circle's diameter.
What is the area of the largest square that fits inside a circle?
The inscribed square's area is A = s² = 2r² = D²/2. For example, a circle with radius 10 fits a square with area 2 × 10² = 200 square units.

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