How this calculator works
Fitting a square inside a circle makes the circle the outer shape, so the square diagonal equals the circle diameter. The diagonal of a square is its side times the square root of two, which gives a side of radius times the square root of two for a square inscribed in a circle. Fitting a circle inside a square makes the square the outer shape, so the circle diameter equals the square side and the radius is half the side. The equal-area comparisons are different again: they do not nest the shapes, they choose a square and a circle with the same enclosed surface. Matching areas gives a square side of radius times the square root of pi, or a circle radius of side divided by the square root of pi. The area ratio shows how much of the outer shape is filled in the nesting cases.
Formula
Largest square in a circle of radius r: side = r√2, area = 2r². Largest circle in a square of side s: radius = s/2, area = πs²/4. Equal areas: if the circle has radius r, the square side is r√π; if the square has side s, the circle radius is s/√π.
Worked example
A unit circle of radius 1 holds a square of side √2 ≈ 1.4142 and area 2, while the circle area is π ≈ 3.1416, so the square fills about 63.66 percent of the circle. A square of side 2 holds a circle of radius 1 and area π, which is about 78.54 percent of the square area of 4. For equal areas with a circle of radius 1, the square side is √π ≈ 1.7725.
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Frequently asked questions
Why is the square side r√2 rather than r?
The square touches the circle at its four corners, so the distance from the centre to a corner is the radius. That distance is half the square diagonal. Since the diagonal is side times √2, the side is 2r/√2, which simplifies to r√2.
Which shape fills more of the other?
The largest circle in a square fills π/4, about 78.54 percent of the square. The largest square in a circle fills 2/π, about 63.66 percent of the circle. The circle is the more efficient filler in this comparison.
What does equal area mean for these shapes?
It means the square and circle enclose exactly the same surface area without one fitting inside the other. The two equal-area modes convert in either direction, so you can start from a circle or from a square.
Can a square have the same perimeter and area as a circle?
Those are different conditions. This page matches areas only. Matching perimeter as well would require a specific size and is not part of these four comparisons.