Square in a Circle Calculator
Enter a circle radius to find the largest inscribed square, or enter a square side to find the circumscribed circle. Formulas, worked examples, and area ratios explained.
🔲 What is a Square in a Circle?
A square inscribed in a circle is the largest square that fits exactly inside a given circle, with all four corners touching the circle's boundary. The relationship between a square and its circumscribed circle is one of the most fundamental constructions in Euclidean geometry. The diagonal of the inscribed square is always equal to the diameter of the circle, because the diagonal spans the full width of the circle from one vertex to the opposite vertex, passing through the center.
This relationship appears in a wide range of practical situations. In manufacturing, a "square peg in a round hole" problem requires knowing the minimum circular hole diameter that can accept a square cross-section shaft. In woodworking, a craftsperson cutting the largest square tile from a circular disk needs the inscribed square formula. In structural engineering, a square cross-section beam inscribed in a circular tube maximizes the wood volume that can be milled from a cylindrical log. Architects designing square courtyards within circular buildings use this relationship to maximize the usable floor area.
There are two complementary ways to approach the square-circle relationship. The first approach starts with a known circle radius and finds the inscribed square. The second starts with a known square side and finds the circumscribed circle. Both use the same core formula: side = r times sqrt(2), rearranged as r = side times sqrt(2) / 2 for the reverse. The area ratio between square and circle is always 2 divided by pi = 63.66%, an elegant constant that holds for any size.
A key theorem underlying this calculator is that every square is a cyclic polygon, meaning all four of its vertices lie on a common circle. The center of this circumscribed circle is the intersection of the diagonals of the square. The diagonals of a square bisect each other at right angles, and each diagonal is a diameter of the circumscribed circle. By Thales' theorem, any angle inscribed in a semicircle and subtending the diameter is a right angle, confirming that the four corners of the square each make a 90-degree angle when viewed from opposite vertices across the diagonal.
📐 Formulas
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Largest Square Tile from a Circular Disk (r = 15 cm)
A woodworker has a circular disk with radius 15 cm and wants the largest square piece that can be cut from it.
Example 2 - Minimum Round Hole for a Square Peg (s = 20 mm)
An engineer needs the minimum circular hole diameter that will pass a square cross-section peg with side 20 mm.
Example 3 - Square Courtyard in a Circular Garden (r = 25 m)
A landscape designer has a circular garden of radius 25 m and wants to inscribe a square paved courtyard.
Example 4 - Circular Log to Square Timber (s = 150 mm)
A sawmill wants a 150 mm by 150 mm square timber beam. Find the minimum circular log diameter needed.
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for a square inscribed in a circle?
For a square inscribed in a circle of radius r: side = r times sqrt(2). This follows from the diagonal of the square equaling the diameter (2r). Since the diagonal of a square with side s is s times sqrt(2), setting s sqrt(2) = 2r gives s = r sqrt(2). Area = 2r squared, perimeter = 4r sqrt(2).
How do you find the circumscribed circle radius of a square?
For a square with side s, the circumscribed circle radius is r = s sqrt(2) / 2. The diagonal of the square (s sqrt(2)) equals the diameter of the circumscribed circle. For side 10: r = 10 times 1.414 / 2 = 7.071 units.
What percentage of a circle is filled by its inscribed square?
The inscribed square fills exactly 2 / pi = 63.66% of the circle area. This is constant regardless of circle size. Circle area = pi r squared. Square area = 2r squared. Ratio = 2r squared / (pi r squared) = 2 / pi = 63.66%.
What is the diagonal of a square inscribed in a circle?
The diagonal of the inscribed square is always equal to the diameter of the circle: diagonal = 2r. Each diagonal of the square passes through the center and connects two points on the circle, making it a diameter by definition.
What is the side of a square that fits exactly in a circle of radius 5?
For r = 5: side = 5 times sqrt(2) = 7.071 units. Area = 50 sq units. Perimeter = 28.284 units. Diagonal = 10 units (equals circle diameter).
What is the relationship between the areas of a square and its circumscribed circle?
Square area / Circle area = s squared / (pi times (s sqrt(2) / 2) squared) = 2 / pi = 0.6366. The square always occupies 63.66% of its circumscribed circle.
What is the circumscribed circle of a unit square?
A unit square (side = 1) has circumscribed circle radius r = sqrt(2) / 2 = 0.7071 units, diameter = sqrt(2) = 1.4142 units, area = pi times 0.5 = 1.5708 square units, and circumference = pi times sqrt(2) = 4.443 units.
Can any square be inscribed in a circle?
Yes. Every square has a unique circumscribed circle because all four vertices are equidistant from the center. The center of the circumscribed circle is the intersection of the diagonals. Conversely, every circle has exactly one largest inscribed square whose diagonal equals the diameter.