Square in a Circle Calculator
Enter one measurement to solve all four classic square-and-circle geometry problems at once: the largest square that fits inside a circle, the largest circle that fits inside a square, the square with the same area as a circle, and the circle with the same area as a square. Results update as you type, with a full worked-step panel and a shape diagram.
Formula
Worked example
Circle with radius 10 cm: inscribed square side = 10 x sqrt(2) = 14.142 cm; square area = 200 cm2; circle area = 314.159 cm2; coverage = 63.66%. Inscribed circle in that same square: r = 14.142 / 2 = 7.071 cm.
What is a square inscribed in a circle?
A square is inscribed in a circle when all four of its corners (vertices) lie exactly on the circle's circumference. The circle is then called the circumscribed circle, or circumcircle, of the square. Because the square's diagonal connects two opposite corners, and both corners are on the circle, the diagonal equals the diameter of the circle. That single relationship, diagonal = 2r, is the foundation of every formula on this page. The largest square that fits inside a given circle is always an inscribed square in this sense: any square with a shorter diagonal would leave unused space, and any square with a longer diagonal would stick outside the circle.
How to calculate the side length of the inscribed square
Start from the Pythagorean theorem. An inscribed square has two diagonals, each equal to the circle diameter 2r. In a square, both diagonals are equal and bisect each other at right angles, so the side s and the diagonal d are related by d^2 = s^2 + s^2 = 2s^2, giving d = s * sqrt(2). Rearranging: s = d / sqrt(2) = 2r / sqrt(2) = r * sqrt(2). For a circle of radius 10 cm, s = 10 * 1.41421... = 14.1421 cm. The square's area is s^2 = 2r^2, and it always covers exactly 2/pi (about 63.66%) of the circle's area, regardless of how big the circle is.
Circle inscribed in a square
The reverse problem - fitting the largest circle inside a square - is simpler. The circle must touch all four sides without crossing any of them, so its diameter equals the side length of the square: d = s, which means r = s / 2. The inscribed circle covers pi/4 of the square's area (about 78.54%), also a constant ratio. The four corner regions, called squircle corners or circular spandrels, together fill the remaining 21.46% of the square. This geometry appears in everything from manhole covers (a circle inside a square frame) to coin blanks and architectural column bases.
Equal-area problems: squaring the circle and circling the square
The classic problem of "squaring the circle" asks for a square that has the same area as a given circle. Since the circle's area is pi * r^2, the square's side must satisfy s^2 = pi * r^2, giving s = r * sqrt(pi) approximately 1.7725 * r. The reverse - a circle with the same area as a given square - requires pi * r^2 = s^2, so r = s / sqrt(pi) approximately 0.5642 * s. Ancient mathematicians sought exact compass-and-straightedge constructions for these equivalences, but it was proved in 1882 (Lindemann-Weierstrass theorem) that pi is transcendental, making the exact classical construction impossible. Numerically, however, the calculation is straightforward.
Key formulas: square and circle relationships
| Problem | Formula | Fixed ratio |
|---|---|---|
| Largest square in circle | s = r√2, A_sq = 2r² | A_sq / A_circ = 2/π ≈ 63.66% |
| Largest circle in square | r = s/2, d = s | A_circ / A_sq = π/4 ≈ 78.54% |
| Equal-area square (from circle) | s = r√π ≈ 1.7725r | A_sq = A_circ |
| Equal-area circle (from square) | r = s/√π ≈ 0.5642s | A_circ = A_sq |
| Square diagonal (any square) | d = s√2 | d/s = √2 ≈ 1.4142 |
| Circumscribed circle of a square | r = s√2/2 = d/2 | r = half-diagonal |
All relationships assume a perfect Euclidean square and circle with no gap or overlap at the boundary.
Frequently asked questions
What is the formula for the largest square that fits inside a circle?
The side length is s = r * sqrt(2), where r is the circle's radius. The square's diagonal equals the circle's diameter (2r), so by the Pythagorean theorem s = d / sqrt(2) = r * sqrt(2). For example, a circle with radius 5 cm has an inscribed square with side 5 * 1.41421 = 7.071 cm and area 50 cm2.
Why does the square's diagonal equal the circle's diameter?
When a square is inscribed in a circle, all four corners touch the circle. The diagonal connects two opposite corners, both on the circle, so the diagonal is a chord that passes through the center, which is the definition of a diameter. Therefore diagonal = 2r.
What percentage of the circle's area does the inscribed square fill?
Always exactly 2/pi, approximately 63.66%, for any circle size. This comes from dividing the square's area (2r^2) by the circle's area (pi * r^2): 2r^2 / (pi * r^2) = 2/pi.
What percentage of a square's area does the inscribed circle fill?
Always pi/4, approximately 78.54%. The inscribed circle has radius r = s/2, so its area is pi * (s/2)^2 = pi * s^2 / 4. Dividing by the square's area s^2 gives pi/4.
How do I find a square with the same area as a circle?
Set s^2 = pi * r^2 and solve: s = r * sqrt(pi). For a circle with radius 6 cm, s = 6 * sqrt(pi) = 6 * 1.7725 = 10.635 cm. The resulting square has area pi * 36 = 113.097 cm2, the same as the circle.
Can you inscribe a square in a circle using only a compass and straightedge?
Yes. Draw a diameter, then draw a perpendicular diameter through the center. The four endpoints of the two diameters are the corners of the inscribed square. This is one of the oldest known geometric constructions.
How is this useful in real life?
Architects use inscribed squares to find the largest rectangular room that fits a circular floor plan. Engineers use them for cutting square stock from round bar stock to minimize waste. The inscribed-circle problem appears in packaging (fitting a round container in a square box), pipe sizing, and coin production.